Theorem frgrwopreg 30177

Description: In a friendship graph there are either no vertices (𝐴 = ∅) or exactly one vertex ((♯‘𝐴) = 1) having degree 𝐾, or all (𝐵 = ∅) or all except one vertices ((♯‘𝐵) = 1) have degree 𝐾. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 3-Jan-2022.)

Hypotheses

Ref	Expression
frgrwopreg.v	⊢ 𝑉 = (Vtx‘𝐺)
frgrwopreg.d	⊢ 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a	⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
frgrwopreg.b	⊢ 𝐵 = (𝑉 ∖ 𝐴)

Assertion

Ref	Expression
frgrwopreg	⊢ (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))

Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝑥,𝐵

Proof of Theorem frgrwopreg

Dummy variables 𝑎 𝑏 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgrwopreg.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		frgrwopreg.d	. . 3 ⊢ 𝐷 = (VtxDeg‘𝐺)
3		frgrwopreg.a	. . 3 ⊢ 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾}
4		frgrwopreg.b	. . 3 ⊢ 𝐵 = (𝑉 ∖ 𝐴)
5	1, 2, 3, 4	frgrwopreglem1 30166	. 2 ⊢ (𝐴 ∈ V ∧ 𝐵 ∈ V)
6		hashv01gt1 14336	. . . 4 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
7		hasheq0 14354	. . . . . 6 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
8		biidd 261	. . . . . 6 ⊢ (𝐴 ∈ V → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = 1))
9		biidd 261	. . . . . 6 ⊢ (𝐴 ∈ V → (1 < (♯‘𝐴) ↔ 1 < (♯‘𝐴)))
10	7, 8, 9	3orbi123d 1431	. . . . 5 ⊢ (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) ↔ (𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴))))
11		hashv01gt1 14336	. . . . . . 7 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)))
12		hasheq0 14354	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 0 ↔ 𝐵 = ∅))
13		biidd 261	. . . . . . . . 9 ⊢ (𝐵 ∈ V → ((♯‘𝐵) = 1 ↔ (♯‘𝐵) = 1))
14		biidd 261	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (1 < (♯‘𝐵) ↔ 1 < (♯‘𝐵)))
15	12, 13, 14	3orbi123d 1431	. . . . . . . 8 ⊢ (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) ↔ (𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵))))
16		olc 866	. . . . . . . . . . 11 ⊢ (𝐵 = ∅ → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
17	16	olcd 872	. . . . . . . . . 10 ⊢ (𝐵 = ∅ → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
18	17	2a1d 26	. . . . . . . . 9 ⊢ (𝐵 = ∅ → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
19		orc 865	. . . . . . . . . . 11 ⊢ ((♯‘𝐵) = 1 → ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))
20	19	olcd 872	. . . . . . . . . 10 ⊢ ((♯‘𝐵) = 1 → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
21	20	2a1d 26	. . . . . . . . 9 ⊢ ((♯‘𝐵) = 1 → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
22		olc 866	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
23	22	orcd 871	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
24	23	2a1d 26	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
25		orc 865	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) = 1 → ((♯‘𝐴) = 1 ∨ 𝐴 = ∅))
26	25	orcd 871	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) = 1 → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
27	26	2a1d 26	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) = 1 → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
28		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
29	1, 2, 3, 4, 28	frgrwopreglem5 30175	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))))
30		frgrusgr 30115	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph)
31		simplll 773	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝐺 ∈ USGraph)
32		elrabi 3668	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑎 ∈ 𝑉)
33	32, 3	eleq2s 2843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑉)
34	33	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑎 ∈ 𝑉)
35	34	ad3antlr 729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑎 ∈ 𝑉)
36		rabidim1 3441	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} → 𝑥 ∈ 𝑉)
37	36, 3	eleq2s 2843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑉)
38	37	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝑉)
39	38	ad3antlr 729	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑥 ∈ 𝑉)
40		simprl 769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑎 ≠ 𝑥)
41		eldifi 4119	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 ∈ (𝑉 ∖ 𝐴) → 𝑏 ∈ 𝑉)
42	41, 4	eleq2s 2843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 ∈ 𝐵 → 𝑏 ∈ 𝑉)
43	42	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑏 ∈ 𝑉)
44	43	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑏 ∈ 𝑉)
45		eldifi 4119	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ (𝑉 ∖ 𝐴) → 𝑦 ∈ 𝑉)
46	45, 4	eleq2s 2843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑉)
47	46	adantl 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝑉)
48	47	ad2antlr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑦 ∈ 𝑉)
49		simprr 771	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → 𝑏 ≠ 𝑦)
50	1, 28	4cyclusnfrgr 30146	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝑉 ∧ 𝑥 ∈ 𝑉 ∧ 𝑎 ≠ 𝑥) ∧ (𝑏 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑏 ≠ 𝑦)) → ((({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → 𝐺 ∉ FriendGraph ))
51	31, 35, 39, 40, 44, 48, 49, 50	syl133anc 1390	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦)) → ((({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → 𝐺 ∉ FriendGraph ))
52	51	exp4b 429	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) → (({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) → (({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺)) → 𝐺 ∉ FriendGraph ))))
53	52	3impd 1345	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → 𝐺 ∉ FriendGraph ))
54		df-nel 3037	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph )
55		pm2.21 123	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝐺 ∈ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
56	54, 55	sylbi 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐺 ∉ FriendGraph → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
57	53, 56	syl6 35	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
58	57	rexlimdvva 3202	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 ∈ USGraph ∧ (𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)) → (∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
59	58	rexlimdvva 3202	. . . . . . . . . . . . . . . . 17 ⊢ (𝐺 ∈ USGraph → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
60	59	com23 86	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ USGraph → (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
61	30, 60	mpcom 38	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
62	61	3ad2ant1 1130	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → (∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ (Edg‘𝐺) ∧ {𝑏, 𝑥} ∈ (Edg‘𝐺)) ∧ ({𝑥, 𝑦} ∈ (Edg‘𝐺) ∧ {𝑦, 𝑎} ∈ (Edg‘𝐺))) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
63	29, 62	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))
64	63	3exp 1116	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ FriendGraph → (1 < (♯‘𝐴) → (1 < (♯‘𝐵) → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
65	64	com3l 89	. . . . . . . . . . 11 ⊢ (1 < (♯‘𝐴) → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
66	24, 27, 65	3jaoi 1424	. . . . . . . . . 10 ⊢ ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (1 < (♯‘𝐵) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
67	66	com12 32	. . . . . . . . 9 ⊢ (1 < (♯‘𝐵) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
68	18, 21, 67	3jaoi 1424	. . . . . . . 8 ⊢ ((𝐵 = ∅ ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
69	15, 68	biimtrdi 252	. . . . . . 7 ⊢ (𝐵 ∈ V → (((♯‘𝐵) = 0 ∨ (♯‘𝐵) = 1 ∨ 1 < (♯‘𝐵)) → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))))
70	11, 69	mpd 15	. . . . . 6 ⊢ (𝐵 ∈ V → ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
71	70	com12 32	. . . . 5 ⊢ ((𝐴 = ∅ ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐵 ∈ V → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
72	10, 71	biimtrdi 252	. . . 4 ⊢ (𝐴 ∈ V → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → (𝐵 ∈ V → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))))
73	6, 72	mpd 15	. . 3 ⊢ (𝐴 ∈ V → (𝐵 ∈ V → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))))
74	73	imp 405	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅))))
75	5, 74	ax-mp 5	1 ⊢ (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∨ wo 845   ∨ w3o 1083   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2930   ∉ wnel 3036  ∃wrex 3060  {crab 3419  Vcvv 3463   ∖ cdif 3936  ∅c0 4318  {cpr 4626   class class class wbr 5143  ‘cfv 6543  0cc0 11138  1c1 11139   < clt 11278  ♯chash 14321  Vtxcvtx 28853  Edgcedg 28904  USGraphcusgr 29006  VtxDegcvtxdg 29323   FriendGraph cfrgr 30112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-n0 12503  df-xnn0 12575  df-z 12589  df-uz 12853  df-xadd 13125  df-fz 13517  df-hash 14322  df-edg 28905  df-uhgr 28915  df-ushgr 28916  df-upgr 28939  df-umgr 28940  df-uspgr 29007  df-usgr 29008  df-nbgr 29190  df-vtxdg 29324  df-frgr 30113

This theorem is referenced by:  frgrregorufr0  30178

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