Theorem nneneq 9228

Description: Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.) Avoid ax-pow 5360. (Revised by BTernaryTau, 11-Nov-2024.)

Assertion

Ref	Expression
nneneq	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem nneneq

Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5146	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑧 ↔ ∅ ≈ 𝑧))
2		eqeq1 2732	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
3	1, 2	imbi12d 344	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (∅ ≈ 𝑧 → ∅ = 𝑧)))
4	3	ralbidv 3173	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)))
5		breq1 5146	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑧 ↔ 𝑦 ≈ 𝑧))
6		eqeq1 2732	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
7	5, 6	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)))
8	7	ralbidv 3173	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)))
9		breq1 5146	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑧 ↔ suc 𝑦 ≈ 𝑧))
10		eqeq1 2732	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 = 𝑧 ↔ suc 𝑦 = 𝑧))
11	9, 10	imbi12d 344	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
12	11	ralbidv 3173	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
13		breq1 5146	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑧 ↔ 𝐴 ≈ 𝑧))
14		eqeq1 2732	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑧 ↔ 𝐴 = 𝑧))
15	13, 14	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (𝐴 ≈ 𝑧 → 𝐴 = 𝑧)))
16	15	ralbidv 3173	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝐴 ≈ 𝑧 → 𝐴 = 𝑧)))
17		0fin 9190	. . . . . . 7 ⊢ ∅ ∈ Fin
18		ensymfib 9206	. . . . . . 7 ⊢ (∅ ∈ Fin → (∅ ≈ 𝑧 ↔ 𝑧 ≈ ∅))
19	17, 18	ax-mp 5	. . . . . 6 ⊢ (∅ ≈ 𝑧 ↔ 𝑧 ≈ ∅)
20		en0 9032	. . . . . . 7 ⊢ (𝑧 ≈ ∅ ↔ 𝑧 = ∅)
21		eqcom 2735	. . . . . . 7 ⊢ (𝑧 = ∅ ↔ ∅ = 𝑧)
22	20, 21	bitri 275	. . . . . 6 ⊢ (𝑧 ≈ ∅ ↔ ∅ = 𝑧)
23	19, 22	sylbb 218	. . . . 5 ⊢ (∅ ≈ 𝑧 → ∅ = 𝑧)
24	23	rgenw 3061	. . . 4 ⊢ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)
25		nn0suc 7896	. . . . . . 7 ⊢ (𝑤 ∈ ω → (𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧))
26		en0 9032	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)
27		breq2 5147	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 ≈ ∅))
28		eqeq2 2740	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = ∅))
29	27, 28	bibi12d 345	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → ((suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)))
30	26, 29	mpbiri 258	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 = 𝑤))
31	30	biimpd 228	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))
32	31	a1i 11	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
33		nfv 1910	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑦 ∈ ω
34		nfra1 3277	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)
35	33, 34	nfan 1895	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧))
36		nfv 1910	. . . . . . . . . 10 ⊢ Ⅎ𝑧(suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)
37		vex 3474	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
38	37	phplem2 9227	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 ≈ suc 𝑧 → 𝑦 ≈ 𝑧))
39	38	imim1d 82	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧)))
40	39	ex 412	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → (𝑧 ∈ ω → ((𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧))))
41	40	a2d 29	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → ((𝑧 ∈ ω → (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧))))
42		rsp 3240	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (𝑧 ∈ ω → (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)))
43	41, 42	impel 505	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧)))
44		suceq 6430	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
45	43, 44	syl8 76	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
46		breq2 5147	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 ≈ suc 𝑧))
47		eqeq2 2740	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = suc 𝑧))
48	46, 47	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑤 = suc 𝑧 → ((suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
49	48	biimprcd 249	. . . . . . . . . . 11 ⊢ ((suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧) → (𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
50	45, 49	syl6 35	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))))
51	35, 36, 50	rexlimd 3259	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (∃𝑧 ∈ ω 𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
52	32, 51	jaod 858	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
53	52	ex 412	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))))
54	25, 53	syl7 74	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (𝑤 ∈ ω → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))))
55	54	ralrimdv 3148	. . . . 5 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → ∀𝑤 ∈ ω (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
56		breq2 5147	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 ≈ 𝑧))
57		eqeq2 2740	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = 𝑧))
58	56, 57	imbi12d 344	. . . . . 6 ⊢ (𝑤 = 𝑧 → ((suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
59	58	cbvralvw 3230	. . . . 5 ⊢ (∀𝑤 ∈ ω (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤) ↔ ∀𝑧 ∈ ω (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧))
60	55, 59	imbitrdi 250	. . . 4 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → ∀𝑧 ∈ ω (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
61	4, 8, 12, 16, 24, 60	finds 7899	. . 3 ⊢ (𝐴 ∈ ω → ∀𝑧 ∈ ω (𝐴 ≈ 𝑧 → 𝐴 = 𝑧))
62		breq2 5147	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 ≈ 𝑧 ↔ 𝐴 ≈ 𝐵))
63		eqeq2 2740	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐵))
64	62, 63	imbi12d 344	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ≈ 𝑧 → 𝐴 = 𝑧) ↔ (𝐴 ≈ 𝐵 → 𝐴 = 𝐵)))
65	64	rspcv 3604	. . 3 ⊢ (𝐵 ∈ ω → (∀𝑧 ∈ ω (𝐴 ≈ 𝑧 → 𝐴 = 𝑧) → (𝐴 ≈ 𝐵 → 𝐴 = 𝐵)))
66	61, 65	mpan9 506	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 → 𝐴 = 𝐵))
67		enrefnn 9066	. . . 4 ⊢ (𝐴 ∈ ω → 𝐴 ≈ 𝐴)
68		breq2 5147	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ≈ 𝐴 ↔ 𝐴 ≈ 𝐵))
69	67, 68	syl5ibcom 244	. . 3 ⊢ (𝐴 ∈ ω → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵))
70	69	adantr 480	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵))
71	66, 70	impbid 211	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099  ∀wral 3057  ∃wrex 3066  ∅c0 4319   class class class wbr 5143  suc csuc 6366  ωcom 7865   ≈ cen 8955  Fincfn 8958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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-om 7866  df-1o 8481  df-en 8959  df-fin 8962

This theorem is referenced by:  php  9229  onomeneqOLD  9248  nnsdomo  9253  fineqvlem  9281  dif1ennnALT  9296  findcard2OLD  9303  cardnn  9981  satfun  35016

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