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Theorem nneneq 6799
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) → (𝐴𝐵𝐴 = 𝐵))

Proof of Theorem nneneq
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3968 . . . . . 6 (𝑥 = ∅ → (𝑥𝑧 ↔ ∅ ≈ 𝑧))
2 eqeq1 2164 . . . . . 6 (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
31, 2imbi12d 233 . . . . 5 (𝑥 = ∅ → ((𝑥𝑧𝑥 = 𝑧) ↔ (∅ ≈ 𝑧 → ∅ = 𝑧)))
43ralbidv 2457 . . . 4 (𝑥 = ∅ → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)))
5 breq1 3968 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
6 eqeq1 2164 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
75, 6imbi12d 233 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝑧𝑥 = 𝑧) ↔ (𝑦𝑧𝑦 = 𝑧)))
87ralbidv 2457 . . . 4 (𝑥 = 𝑦 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)))
9 breq1 3968 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑧 ↔ suc 𝑦𝑧))
10 eqeq1 2164 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥 = 𝑧 ↔ suc 𝑦 = 𝑧))
119, 10imbi12d 233 . . . . 5 (𝑥 = suc 𝑦 → ((𝑥𝑧𝑥 = 𝑧) ↔ (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
1211ralbidv 2457 . . . 4 (𝑥 = suc 𝑦 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
13 breq1 3968 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑧𝐴𝑧))
14 eqeq1 2164 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑧𝐴 = 𝑧))
1513, 14imbi12d 233 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑧𝑥 = 𝑧) ↔ (𝐴𝑧𝐴 = 𝑧)))
1615ralbidv 2457 . . . 4 (𝑥 = 𝐴 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧)))
17 ensym 6723 . . . . . 6 (∅ ≈ 𝑧𝑧 ≈ ∅)
18 en0 6737 . . . . . . 7 (𝑧 ≈ ∅ ↔ 𝑧 = ∅)
19 eqcom 2159 . . . . . . 7 (𝑧 = ∅ ↔ ∅ = 𝑧)
2018, 19bitri 183 . . . . . 6 (𝑧 ≈ ∅ ↔ ∅ = 𝑧)
2117, 20sylib 121 . . . . 5 (∅ ≈ 𝑧 → ∅ = 𝑧)
2221rgenw 2512 . . . 4 𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)
23 nn0suc 4562 . . . . . . 7 (𝑤 ∈ ω → (𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧))
24 en0 6737 . . . . . . . . . . . 12 (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)
25 breq2 3969 . . . . . . . . . . . . 13 (𝑤 = ∅ → (suc 𝑦𝑤 ↔ suc 𝑦 ≈ ∅))
26 eqeq2 2167 . . . . . . . . . . . . 13 (𝑤 = ∅ → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = ∅))
2725, 26bibi12d 234 . . . . . . . . . . . 12 (𝑤 = ∅ → ((suc 𝑦𝑤 ↔ suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)))
2824, 27mpbiri 167 . . . . . . . . . . 11 (𝑤 = ∅ → (suc 𝑦𝑤 ↔ suc 𝑦 = 𝑤))
2928biimpd 143 . . . . . . . . . 10 (𝑤 = ∅ → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))
3029a1i 9 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑤 = ∅ → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
31 nfv 1508 . . . . . . . . . . 11 𝑧 𝑦 ∈ ω
32 nfra1 2488 . . . . . . . . . . 11 𝑧𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)
3331, 32nfan 1545 . . . . . . . . . 10 𝑧(𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧))
34 nfv 1508 . . . . . . . . . 10 𝑧(suc 𝑦𝑤 → suc 𝑦 = 𝑤)
35 rsp 2504 . . . . . . . . . . . . . 14 (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → (𝑧 ∈ ω → (𝑦𝑧𝑦 = 𝑧)))
36 vex 2715 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
37 vex 2715 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
3836, 37phplem4 6797 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 ≈ suc 𝑧𝑦𝑧))
3938imim1d 75 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦𝑧𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧)))
4039ex 114 . . . . . . . . . . . . . . 15 (𝑦 ∈ ω → (𝑧 ∈ ω → ((𝑦𝑧𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧))))
4140a2d 26 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → ((𝑧 ∈ ω → (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧))))
4235, 41syl5 32 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧))))
4342imp 123 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧)))
44 suceq 4362 . . . . . . . . . . . 12 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
4543, 44syl8 71 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
46 breq2 3969 . . . . . . . . . . . . 13 (𝑤 = suc 𝑧 → (suc 𝑦𝑤 ↔ suc 𝑦 ≈ suc 𝑧))
47 eqeq2 2167 . . . . . . . . . . . . 13 (𝑤 = suc 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = suc 𝑧))
4846, 47imbi12d 233 . . . . . . . . . . . 12 (𝑤 = suc 𝑧 → ((suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
4948biimprcd 159 . . . . . . . . . . 11 ((suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧) → (𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5045, 49syl6 33 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))))
5133, 34, 50rexlimd 2571 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (∃𝑧 ∈ ω 𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5230, 51jaod 707 . . . . . . . 8 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5352ex 114 . . . . . . 7 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))))
5423, 53syl7 69 . . . . . 6 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → (𝑤 ∈ ω → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))))
5554ralrimdv 2536 . . . . 5 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ∀𝑤 ∈ ω (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
56 breq2 3969 . . . . . . 7 (𝑤 = 𝑧 → (suc 𝑦𝑤 ↔ suc 𝑦𝑧))
57 eqeq2 2167 . . . . . . 7 (𝑤 = 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = 𝑧))
5856, 57imbi12d 233 . . . . . 6 (𝑤 = 𝑧 → ((suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
5958cbvralv 2680 . . . . 5 (∀𝑤 ∈ ω (suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧))
6055, 59syl6ib 160 . . . 4 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
614, 8, 12, 16, 22, 60finds 4558 . . 3 (𝐴 ∈ ω → ∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧))
62 breq2 3969 . . . . 5 (𝑧 = 𝐵 → (𝐴𝑧𝐴𝐵))
63 eqeq2 2167 . . . . 5 (𝑧 = 𝐵 → (𝐴 = 𝑧𝐴 = 𝐵))
6462, 63imbi12d 233 . . . 4 (𝑧 = 𝐵 → ((𝐴𝑧𝐴 = 𝑧) ↔ (𝐴𝐵𝐴 = 𝐵)))
6564rspcv 2812 . . 3 (𝐵 ∈ ω → (∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧) → (𝐴𝐵𝐴 = 𝐵)))
6661, 65mpan9 279 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
67 eqeng 6708 . . 3 (𝐴 ∈ ω → (𝐴 = 𝐵𝐴𝐵))
6867adantr 274 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
6966, 68impbid 128 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 698   = wceq 1335  wcel 2128  wral 2435  wrex 2436  c0 3394   class class class wbr 3965  suc csuc 4325  ωcom 4548  cen 6680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-er 6477  df-en 6683
This theorem is referenced by:  findcard2  6831  findcard2s  6832  unsnfidcex  6861  unsnfidcel  6862  exmidonfinlem  7123  hashen  10653  hashunlem  10673
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