dfsclnbgr6 - Mathbox for Alexander van der Vekens

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Theorem dfsclnbgr6 47330

Description: Alternate definition of a semiclosed neighborhood of a vertex as a union of a semiopen neighborhood and the vertex itself if there is a loop at this vertex. (Contributed by AV, 17-May-2025.)

**Hypotheses**
Ref	Expression
dfvopnbgr2.v	⊢ 𝑉 = (Vtx‘𝐺)
dfvopnbgr2.e	⊢ 𝐸 = (Edg‘𝐺)
dfvopnbgr2.u	⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
dfsclnbgr6.s	⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}

**Assertion**
Ref	Expression
dfsclnbgr6	⊢ (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))

Distinct variable groups: 𝑒,𝐸 𝑒,𝐺 𝑒,𝑁,𝑛 𝑒,𝑉,𝑛 𝑛,𝐸 𝑛,𝐺 Allowed substitution hints: 𝑆(𝑒,𝑛) 𝑈(𝑒,𝑛)

**Proof of Theorem dfsclnbgr6**
Step	Hyp	Ref	Expression
1		simpr 483	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → 𝑛 ∈ 𝑒)
2	1	anim1i 613	. . . . . . . . . 10 ⊢ (((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∧ 𝑛 = 𝑁) → (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))
3	2	olcd 872	. . . . . . . . 9 ⊢ (((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∧ 𝑛 = 𝑁) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)))
4	3	expcom 412	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
5		3anass 1092	. . . . . . . . . . . 12 ⊢ ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
6	5	biimpri 227	. . . . . . . . . . 11 ⊢ ((𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)) → (𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))
7	6	orcd 871	. . . . . . . . . 10 ⊢ ((𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)) → ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})))
8	7	orcd 871	. . . . . . . . 9 ⊢ ((𝑛 ≠ 𝑁 ∧ (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)))
9	8	ex 411	. . . . . . . 8 ⊢ (𝑛 ≠ 𝑁 → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
10	4, 9	pm2.61ine 3014	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)))
11		3simpc 1147	. . . . . . . . . 10 ⊢ ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))
12	11	a1i 11	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑉 → ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
13		vsnid 4667	. . . . . . . . . . . . . . . 16 ⊢ 𝑛 ∈ {𝑛}
14	13	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = {𝑛} → 𝑛 ∈ {𝑛})
15		eleq2 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑒 = {𝑛} → (𝑛 ∈ 𝑒 ↔ 𝑛 ∈ {𝑛}))
16	14, 15	mpbird 256	. . . . . . . . . . . . . 14 ⊢ (𝑒 = {𝑛} → 𝑛 ∈ 𝑒)
17	16	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) → 𝑛 ∈ 𝑒)
18		eleq1 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ 𝑒 ↔ 𝑁 ∈ 𝑒))
19	18	bicomd 222	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → (𝑁 ∈ 𝑒 ↔ 𝑛 ∈ 𝑒))
20	19	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) → (𝑁 ∈ 𝑒 ↔ 𝑛 ∈ 𝑒))
21	17, 20	mpbird 256	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) → 𝑁 ∈ 𝑒)
22	21	adantl 480	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) → 𝑁 ∈ 𝑒)
23	17	adantl 480	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) → 𝑛 ∈ 𝑒)
24	22, 23	jca 510	. . . . . . . . . 10 ⊢ ((𝑁 ∈ 𝑉 ∧ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))
25	24	ex 411	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑉 → ((𝑛 = 𝑁 ∧ 𝑒 = {𝑛}) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
26	12, 25	jaod 857	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
27	18	biimpac 477	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) → 𝑁 ∈ 𝑒)
28		simpl 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) → 𝑛 ∈ 𝑒)
29	27, 28	jca 510	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒))
30	29	a1i 11	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑉 → ((𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
31	26, 30	jaod 857	. . . . . . 7 ⊢ (𝑁 ∈ 𝑉 → ((((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)) → (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)))
32	10, 31	impbid2 225	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → ((𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
33	32	rexbidv 3168	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ ∃𝑒 ∈ 𝐸 (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
34		r19.43 3111	. . . . . 6 ⊢ (∃𝑒 ∈ 𝐸 (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)) ↔ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ ∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)))
35	34	a1i 11	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 (((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)) ↔ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ ∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))))
36		r19.41v 3178	. . . . . . . 8 ⊢ (∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) ↔ (∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁))
37	36	biancomi 461	. . . . . . 7 ⊢ (∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) ↔ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))
38	37	a1i 11	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁) ↔ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)))
39	38	orbi2d 913	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → ((∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ ∃𝑒 ∈ 𝐸 (𝑛 ∈ 𝑒 ∧ 𝑛 = 𝑁)) ↔ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))))
40	33, 35, 39	3bitrd 304	. . . 4 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ↔ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))))
41	40	rabbidv 3426	. . 3 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = {𝑛 ∈ 𝑉 ∣ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))})
42		unrab 4304	. . . 4 ⊢ ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ 𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)}) = {𝑛 ∈ 𝑉 ∣ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))}
43		rabsneq 4648	. . . . . 6 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒} = {𝑛 ∈ 𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)})
44	43	eqcomd 2731	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)} = {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒})
45	44	uneq2d 4160	. . . 4 ⊢ (𝑁 ∈ 𝑉 → ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ 𝑉 ∣ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒)}) = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
46	42, 45	eqtr3id 2779	. . 3 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ (∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛})) ∨ (𝑛 = 𝑁 ∧ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒))} = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
47	41, 46	eqtrd 2765	. 2 ⊢ (𝑁 ∈ 𝑉 → {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)} = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
48		dfvopnbgr2.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
49		dfsclnbgr6.s	. . 3 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
50		dfvopnbgr2.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
51	48, 49, 50	dfsclnbgr2 47318	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 (𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒)})
52		dfvopnbgr2.u	. . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
53	48, 50, 52	dfvopnbgr2 47325	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑈 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))})
54	53	uneq1d 4159	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}) = ({𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 ((𝑛 ≠ 𝑁 ∧ 𝑁 ∈ 𝑒 ∧ 𝑛 ∈ 𝑒) ∨ (𝑛 = 𝑁 ∧ 𝑒 = {𝑛}))} ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
55	47, 51, 54	3eqtr4d 2775	1 ⊢ (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 {crab 3418 ∪ cun 3942 ⊆ wss 3944 {csn 4630 {cpr 4632 ‘cfv 6549 (class class class)co 7419 Vtxcvtx 28881 Edgcedg 28932 NeighbVtx cnbgr 29217

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-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-nbgr 29218

This theorem is referenced by: dfnbgrss2 47331

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