Theorem nneneq 6851

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 4003	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ≈ 𝑧 ↔ ∅ ≈ 𝑧))
2		eqeq1 2184	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
3	1, 2	imbi12d 234	. . . . 5 ⊢ (𝑥 = ∅ → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (∅ ≈ 𝑧 → ∅ = 𝑧)))
4	3	ralbidv 2477	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)))
5		breq1 4003	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑧 ↔ 𝑦 ≈ 𝑧))
6		eqeq1 2184	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 ↔ 𝑦 = 𝑧))
7	5, 6	imbi12d 234	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)))
8	7	ralbidv 2477	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)))
9		breq1 4003	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 ≈ 𝑧 ↔ suc 𝑦 ≈ 𝑧))
10		eqeq1 2184	. . . . . 6 ⊢ (𝑥 = suc 𝑦 → (𝑥 = 𝑧 ↔ suc 𝑦 = 𝑧))
11	9, 10	imbi12d 234	. . . . 5 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
12	11	ralbidv 2477	. . . 4 ⊢ (𝑥 = suc 𝑦 → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
13		breq1 4003	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑧 ↔ 𝐴 ≈ 𝑧))
14		eqeq1 2184	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝑧 ↔ 𝐴 = 𝑧))
15	13, 14	imbi12d 234	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ (𝐴 ≈ 𝑧 → 𝐴 = 𝑧)))
16	15	ralbidv 2477	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑧 ∈ ω (𝑥 ≈ 𝑧 → 𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝐴 ≈ 𝑧 → 𝐴 = 𝑧)))
17		ensym 6775	. . . . . 6 ⊢ (∅ ≈ 𝑧 → 𝑧 ≈ ∅)
18		en0 6789	. . . . . . 7 ⊢ (𝑧 ≈ ∅ ↔ 𝑧 = ∅)
19		eqcom 2179	. . . . . . 7 ⊢ (𝑧 = ∅ ↔ ∅ = 𝑧)
20	18, 19	bitri 184	. . . . . 6 ⊢ (𝑧 ≈ ∅ ↔ ∅ = 𝑧)
21	17, 20	sylib 122	. . . . 5 ⊢ (∅ ≈ 𝑧 → ∅ = 𝑧)
22	21	rgenw 2532	. . . 4 ⊢ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)
23		nn0suc 4600	. . . . . . 7 ⊢ (𝑤 ∈ ω → (𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧))
24		en0 6789	. . . . . . . . . . . 12 ⊢ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)
25		breq2 4004	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 ≈ ∅))
26		eqeq2 2187	. . . . . . . . . . . . 13 ⊢ (𝑤 = ∅ → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = ∅))
27	25, 26	bibi12d 235	. . . . . . . . . . . 12 ⊢ (𝑤 = ∅ → ((suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)))
28	24, 27	mpbiri 168	. . . . . . . . . . 11 ⊢ (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 = 𝑤))
29	28	biimpd 144	. . . . . . . . . 10 ⊢ (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))
30	29	a1i 9	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑤 = ∅ → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
31		nfv 1528	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑦 ∈ ω
32		nfra1 2508	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)
33	31, 32	nfan 1565	. . . . . . . . . 10 ⊢ Ⅎ𝑧(𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧))
34		nfv 1528	. . . . . . . . . 10 ⊢ Ⅎ𝑧(suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)
35		rsp 2524	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (𝑧 ∈ ω → (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)))
36		vex 2740	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
37		vex 2740	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
38	36, 37	phplem4 6849	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 ≈ suc 𝑧 → 𝑦 ≈ 𝑧))
39	38	imim1d 75	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧)))
40	39	ex 115	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ω → (𝑧 ∈ ω → ((𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧))))
41	40	a2d 26	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ω → ((𝑧 ∈ ω → (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧))))
42	35, 41	syl5 32	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧))))
43	42	imp 124	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → 𝑦 = 𝑧)))
44		suceq 4399	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
45	43, 44	syl8 71	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
46		breq2 4004	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 ≈ suc 𝑧))
47		eqeq2 2187	. . . . . . . . . . . . 13 ⊢ (𝑤 = suc 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = suc 𝑧))
48	46, 47	imbi12d 234	. . . . . . . . . . . 12 ⊢ (𝑤 = suc 𝑧 → ((suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
49	48	biimprcd 160	. . . . . . . . . . 11 ⊢ ((suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧) → (𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
50	45, 49	syl6 33	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (𝑧 ∈ ω → (𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))))
51	33, 34, 50	rexlimd 2591	. . . . . . . . 9 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → (∃𝑧 ∈ ω 𝑤 = suc 𝑧 → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
52	30, 51	jaod 717	. . . . . . . 8 ⊢ ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧)) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
53	52	ex 115	. . . . . . 7 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))))
54	23, 53	syl7 69	. . . . . 6 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → (𝑤 ∈ ω → (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤))))
55	54	ralrimdv 2556	. . . . 5 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → ∀𝑤 ∈ ω (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤)))
56		breq2 4004	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (suc 𝑦 ≈ 𝑤 ↔ suc 𝑦 ≈ 𝑧))
57		eqeq2 2187	. . . . . . 7 ⊢ (𝑤 = 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = 𝑧))
58	56, 57	imbi12d 234	. . . . . 6 ⊢ (𝑤 = 𝑧 → ((suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
59	58	cbvralv 2703	. . . . 5 ⊢ (∀𝑤 ∈ ω (suc 𝑦 ≈ 𝑤 → suc 𝑦 = 𝑤) ↔ ∀𝑧 ∈ ω (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧))
60	55, 59	syl6ib 161	. . . 4 ⊢ (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦 ≈ 𝑧 → 𝑦 = 𝑧) → ∀𝑧 ∈ ω (suc 𝑦 ≈ 𝑧 → suc 𝑦 = 𝑧)))
61	4, 8, 12, 16, 22, 60	finds 4596	. . 3 ⊢ (𝐴 ∈ ω → ∀𝑧 ∈ ω (𝐴 ≈ 𝑧 → 𝐴 = 𝑧))
62		breq2 4004	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 ≈ 𝑧 ↔ 𝐴 ≈ 𝐵))
63		eqeq2 2187	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐵))
64	62, 63	imbi12d 234	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝐴 ≈ 𝑧 → 𝐴 = 𝑧) ↔ (𝐴 ≈ 𝐵 → 𝐴 = 𝐵)))
65	64	rspcv 2837	. . 3 ⊢ (𝐵 ∈ ω → (∀𝑧 ∈ ω (𝐴 ≈ 𝑧 → 𝐴 = 𝑧) → (𝐴 ≈ 𝐵 → 𝐴 = 𝐵)))
66	61, 65	mpan9 281	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 → 𝐴 = 𝐵))
67		eqeng 6760	. . 3 ⊢ (𝐴 ∈ ω → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵))
68	67	adantr 276	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵 → 𝐴 ≈ 𝐵))
69	66, 68	impbid 129	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ≈ 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  ∅c0 3422   class class class wbr 4000  suc csuc 4362  ωcom 4586   ≈ cen 6732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-er 6529  df-en 6735

This theorem is referenced by:  findcard2  6883  findcard2s  6884  unsnfidcex  6913  unsnfidcel  6914  exmidonfinlem  7186  hashen  10748  hashunlem  10768