Theorem nneneq 8694

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.)

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 5061	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑧 ↔ ∅ ≈ 𝑧))
2		eqeq1 2825	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
3	1, 2	imbi12d 347	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (∅ ≈ 𝑧 → ∅ = 𝑧)))
4	3	ralbidv 3197	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)))
5		breq1 5061	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑧 ↔ 𝑦 ≈ 𝑧))
6		eqeq1 2825	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
7	5, 6	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)))
8	7	ralbidv 3197	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)))
9		breq1 5061	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑧 ↔ suc 𝑦 ≈ 𝑧))
10		eqeq1 2825	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 = 𝑧 ↔ suc 𝑦 = 𝑧))
11	9, 10	imbi12d 347	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
12	11	ralbidv 3197	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
13		breq1 5061	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑧 ↔ 𝐴 ≈ 𝑧))
14		eqeq1 2825	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑧 ↔ 𝐴 = 𝑧))
15	13, 14	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (𝐴 ≈ 𝑧 → 𝐴 = 𝑧)))
16	15	ralbidv 3197	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝐴 ≈ 𝑧 → 𝐴 = 𝑧)))
17		ensym 8552	. . . . . 6 ⊢ (∅ ≈ 𝑧 → 𝑧 ≈ ∅)
18		en0 8566	. . . . . . 7 ⊢ (𝑧 ≈ ∅ ↔ 𝑧 = ∅)
19		eqcom 2828	. . . . . . 7 ⊢ (𝑧 = ∅ ↔ ∅ = 𝑧)
20	18, 19	bitri 277	. . . . . 6 ⊢ (𝑧 ≈ ∅ ↔ ∅ = 𝑧)
21	17, 20	sylib 220	. . . . 5 ⊢ (∅ ≈ 𝑧 → ∅ = 𝑧)
22	21	rgenw 3150	. . . 4 ⊢ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)
23		nn0suc 7600	. . . . . . 7 ⊢ (𝑤 ∈ ω → (𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧))
24		en0 8566	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)
25		breq2 5062	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 ≈ ∅))
26		eqeq2 2833	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = ∅))
27	25, 26	bibi12d 348	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → ((suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)))
28	24, 27	mpbiri 260	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 = 𝑤))
29	28	biimpd 231	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))
30	29	a1i 11	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
31		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑦 ∈ ω
32		nfra1 3219	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)
33	31, 32	nfan 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧))
34		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑧(suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)
35		vex 3497	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
36		vex 3497	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
37	35, 36	phplem4 8693	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 ≈ suc 𝑧 → 𝑦 ≈ 𝑧))
38	37	imim1d 82	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧)))
39	38	ex 415	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → (𝑧 ∈ ω → ((𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧))))
40	39	a2d 29	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → ((𝑧 ∈ ω → (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧))))
41		rsp 3205	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (𝑧 ∈ ω → (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)))
42	40, 41	impel 508	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧)))
43		suceq 6250	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
44	42, 43	syl8 76	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
45		breq2 5062	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 ≈ suc 𝑧))
46		eqeq2 2833	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = suc 𝑧))
47	45, 46	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑤 = suc 𝑧 → ((suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
48	47	biimprcd 252	. . . . . . . . . . 11 ⊢ ((suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧) → (𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
49	44, 48	syl6 35	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))))
50	33, 34, 49	rexlimd 3317	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (∃𝑧 ∈ ω 𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
51	30, 50	jaod 855	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
52	51	ex 415	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))))
53	23, 52	syl7 74	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (𝑤 ∈ ω → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))))
54	53	ralrimdv 3188	. . . . 5 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → ∀𝑤 ∈ ω (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
55		breq2 5062	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 ≈ 𝑧))
56		eqeq2 2833	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = 𝑧))
57	55, 56	imbi12d 347	. . . . . 6 ⊢ (𝑤 = 𝑧 → ((suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
58	57	cbvralvw 3449	. . . . 5 ⊢ (∀𝑤 ∈ ω (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤) ↔ ∀𝑧 ∈ ω (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧))
59	54, 58	syl6ib 253	. . . 4 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → ∀𝑧 ∈ ω (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
60	4, 8, 12, 16, 22, 59	finds 7602	. . 3 ⊢ (𝐴 ∈ ω → ∀𝑧 ∈ ω (𝐴 ≈ 𝑧 → 𝐴 = 𝑧))
61		breq2 5062	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 ≈ 𝑧 ↔ 𝐴 ≈ 𝐵))
62		eqeq2 2833	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐵))
63	61, 62	imbi12d 347	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ≈ 𝑧 → 𝐴 = 𝑧) ↔ (𝐴 ≈ 𝐵 → 𝐴 = 𝐵)))
64	63	rspcv 3617	. . 3 ⊢ (𝐵 ∈ ω → (∀𝑧 ∈ ω (𝐴 ≈ 𝑧 → 𝐴 = 𝑧) → (𝐴 ≈ 𝐵 → 𝐴 = 𝐵)))
65	60, 64	mpan9 509	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 → 𝐴 = 𝐵))
66		eqeng 8537	. . 3 ⊢ (𝐴 ∈ ω → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵))
67	66	adantr 483	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵))
68	65, 67	impbid 214	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  ∅c0 4290   class class class wbr 5058  suc csuc 6187  ωcom 7574   ≈ cen 8500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-om 7575  df-er 8283  df-en 8504

This theorem is referenced by:  php  8695  onomeneq  8702  nnsdomo  8707  fineqvlem  8726  dif1en  8745  findcard2  8752  cardnn  9386  satfun  32653