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Theorem nneneq 8300
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 4799 . . . . . 6 (𝑥 = ∅ → (𝑥𝑧 ↔ ∅ ≈ 𝑧))
2 eqeq1 2756 . . . . . 6 (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
31, 2imbi12d 333 . . . . 5 (𝑥 = ∅ → ((𝑥𝑧𝑥 = 𝑧) ↔ (∅ ≈ 𝑧 → ∅ = 𝑧)))
43ralbidv 3116 . . . 4 (𝑥 = ∅ → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)))
5 breq1 4799 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
6 eqeq1 2756 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
75, 6imbi12d 333 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝑧𝑥 = 𝑧) ↔ (𝑦𝑧𝑦 = 𝑧)))
87ralbidv 3116 . . . 4 (𝑥 = 𝑦 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)))
9 breq1 4799 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑧 ↔ suc 𝑦𝑧))
10 eqeq1 2756 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥 = 𝑧 ↔ suc 𝑦 = 𝑧))
119, 10imbi12d 333 . . . . 5 (𝑥 = suc 𝑦 → ((𝑥𝑧𝑥 = 𝑧) ↔ (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
1211ralbidv 3116 . . . 4 (𝑥 = suc 𝑦 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
13 breq1 4799 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑧𝐴𝑧))
14 eqeq1 2756 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑧𝐴 = 𝑧))
1513, 14imbi12d 333 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑧𝑥 = 𝑧) ↔ (𝐴𝑧𝐴 = 𝑧)))
1615ralbidv 3116 . . . 4 (𝑥 = 𝐴 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧)))
17 ensym 8162 . . . . . 6 (∅ ≈ 𝑧𝑧 ≈ ∅)
18 en0 8176 . . . . . . 7 (𝑧 ≈ ∅ ↔ 𝑧 = ∅)
19 eqcom 2759 . . . . . . 7 (𝑧 = ∅ ↔ ∅ = 𝑧)
2018, 19bitri 264 . . . . . 6 (𝑧 ≈ ∅ ↔ ∅ = 𝑧)
2117, 20sylib 208 . . . . 5 (∅ ≈ 𝑧 → ∅ = 𝑧)
2221rgenw 3054 . . . 4 𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)
23 nn0suc 7247 . . . . . . 7 (𝑤 ∈ ω → (𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧))
24 en0 8176 . . . . . . . . . . . 12 (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)
25 breq2 4800 . . . . . . . . . . . . 13 (𝑤 = ∅ → (suc 𝑦𝑤 ↔ suc 𝑦 ≈ ∅))
26 eqeq2 2763 . . . . . . . . . . . . 13 (𝑤 = ∅ → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = ∅))
2725, 26bibi12d 334 . . . . . . . . . . . 12 (𝑤 = ∅ → ((suc 𝑦𝑤 ↔ suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)))
2824, 27mpbiri 248 . . . . . . . . . . 11 (𝑤 = ∅ → (suc 𝑦𝑤 ↔ suc 𝑦 = 𝑤))
2928biimpd 219 . . . . . . . . . 10 (𝑤 = ∅ → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))
3029a1i 11 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑤 = ∅ → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
31 nfv 1984 . . . . . . . . . . 11 𝑧 𝑦 ∈ ω
32 nfra1 3071 . . . . . . . . . . 11 𝑧𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)
3331, 32nfan 1969 . . . . . . . . . 10 𝑧(𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧))
34 nfv 1984 . . . . . . . . . 10 𝑧(suc 𝑦𝑤 → suc 𝑦 = 𝑤)
35 rsp 3059 . . . . . . . . . . . . . 14 (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → (𝑧 ∈ ω → (𝑦𝑧𝑦 = 𝑧)))
36 vex 3335 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
37 vex 3335 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
3836, 37phplem4 8299 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 ≈ suc 𝑧𝑦𝑧))
3938imim1d 82 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦𝑧𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧)))
4039ex 449 . . . . . . . . . . . . . . 15 (𝑦 ∈ ω → (𝑧 ∈ ω → ((𝑦𝑧𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧))))
4140a2d 29 . . . . . . . . . . . . . 14 (𝑦 ∈ ω → ((𝑧 ∈ ω → (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧))))
4235, 41syl5 34 . . . . . . . . . . . . 13 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧))))
4342imp 444 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧)))
44 suceq 5943 . . . . . . . . . . . 12 (𝑦 = 𝑧 → suc 𝑦 = suc 𝑧)
4543, 44syl8 76 . . . . . . . . . . 11 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
46 breq2 4800 . . . . . . . . . . . . 13 (𝑤 = suc 𝑧 → (suc 𝑦𝑤 ↔ suc 𝑦 ≈ suc 𝑧))
47 eqeq2 2763 . . . . . . . . . . . . 13 (𝑤 = suc 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = suc 𝑧))
4846, 47imbi12d 333 . . . . . . . . . . . 12 (𝑤 = suc 𝑧 → ((suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
4948biimprcd 240 . . . . . . . . . . 11 ((suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧) → (𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5045, 49syl6 35 . . . . . . . . . 10 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))))
5133, 34, 50rexlimd 3156 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (∃𝑧 ∈ ω 𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5230, 51jaod 394 . . . . . . . 8 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5352ex 449 . . . . . . 7 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))))
5423, 53syl7 74 . . . . . 6 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → (𝑤 ∈ ω → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))))
5554ralrimdv 3098 . . . . 5 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ∀𝑤 ∈ ω (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
56 breq2 4800 . . . . . . 7 (𝑤 = 𝑧 → (suc 𝑦𝑤 ↔ suc 𝑦𝑧))
57 eqeq2 2763 . . . . . . 7 (𝑤 = 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = 𝑧))
5856, 57imbi12d 333 . . . . . 6 (𝑤 = 𝑧 → ((suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
5958cbvralv 3302 . . . . 5 (∀𝑤 ∈ ω (suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧))
6055, 59syl6ib 241 . . . 4 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
614, 8, 12, 16, 22, 60finds 7249 . . 3 (𝐴 ∈ ω → ∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧))
62 breq2 4800 . . . . 5 (𝑧 = 𝐵 → (𝐴𝑧𝐴𝐵))
63 eqeq2 2763 . . . . 5 (𝑧 = 𝐵 → (𝐴 = 𝑧𝐴 = 𝐵))
6462, 63imbi12d 333 . . . 4 (𝑧 = 𝐵 → ((𝐴𝑧𝐴 = 𝑧) ↔ (𝐴𝐵𝐴 = 𝐵)))
6564rspcv 3437 . . 3 (𝐵 ∈ ω → (∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧) → (𝐴𝐵𝐴 = 𝐵)))
6661, 65mpan9 487 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
67 eqeng 8147 . . 3 (𝐴 ∈ ω → (𝐴 = 𝐵𝐴𝐵))
6867adantr 472 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
6966, 68impbid 202 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1624  wcel 2131  wral 3042  wrex 3043  c0 4050   class class class wbr 4796  suc csuc 5878  ωcom 7222  cen 8110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-er 7903  df-en 8114
This theorem is referenced by:  php  8301  onomeneq  8307  nnsdomo  8312  fineqvlem  8331  dif1en  8350  findcard2  8357  cardnn  8971
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