ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nneneq GIF version

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) → (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 4003 . . . . . 6 (𝑥 = ∅ → (𝑥𝑧 ↔ ∅ ≈ 𝑧))
2 eqeq1 2184 . . . . . 6 (𝑥 = ∅ → (𝑥 = 𝑧 ↔ ∅ = 𝑧))
31, 2imbi12d 234 . . . . 5 (𝑥 = ∅ → ((𝑥𝑧𝑥 = 𝑧) ↔ (∅ ≈ 𝑧 → ∅ = 𝑧)))
43ralbidv 2477 . . . 4 (𝑥 = ∅ → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (∅ ≈ 𝑧 → ∅ = 𝑧)))
5 breq1 4003 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
6 eqeq1 2184 . . . . . 6 (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))
75, 6imbi12d 234 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝑧𝑥 = 𝑧) ↔ (𝑦𝑧𝑦 = 𝑧)))
87ralbidv 2477 . . . 4 (𝑥 = 𝑦 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)))
9 breq1 4003 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥𝑧 ↔ suc 𝑦𝑧))
10 eqeq1 2184 . . . . . 6 (𝑥 = suc 𝑦 → (𝑥 = 𝑧 ↔ suc 𝑦 = 𝑧))
119, 10imbi12d 234 . . . . 5 (𝑥 = suc 𝑦 → ((𝑥𝑧𝑥 = 𝑧) ↔ (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
1211ralbidv 2477 . . . 4 (𝑥 = suc 𝑦 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
13 breq1 4003 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑧𝐴𝑧))
14 eqeq1 2184 . . . . . 6 (𝑥 = 𝐴 → (𝑥 = 𝑧𝐴 = 𝑧))
1513, 14imbi12d 234 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝑧𝑥 = 𝑧) ↔ (𝐴𝑧𝐴 = 𝑧)))
1615ralbidv 2477 . . . 4 (𝑥 = 𝐴 → (∀𝑧 ∈ ω (𝑥𝑧𝑥 = 𝑧) ↔ ∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧)))
17 ensym 6775 . . . . . 6 (∅ ≈ 𝑧𝑧 ≈ ∅)
18 en0 6789 . . . . . . 7 (𝑧 ≈ ∅ ↔ 𝑧 = ∅)
19 eqcom 2179 . . . . . . 7 (𝑧 = ∅ ↔ ∅ = 𝑧)
2018, 19bitri 184 . . . . . 6 (𝑧 ≈ ∅ ↔ ∅ = 𝑧)
2117, 20sylib 122 . . . . 5 (∅ ≈ 𝑧 → ∅ = 𝑧)
2221rgenw 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 𝑦𝑤 ↔ suc 𝑦 ≈ ∅))
26 eqeq2 2187 . . . . . . . . . . . . 13 (𝑤 = ∅ → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = ∅))
2725, 26bibi12d 235 . . . . . . . . . . . 12 (𝑤 = ∅ → ((suc 𝑦𝑤 ↔ suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ ∅ ↔ suc 𝑦 = ∅)))
2824, 27mpbiri 168 . . . . . . . . . . 11 (𝑤 = ∅ → (suc 𝑦𝑤 ↔ suc 𝑦 = 𝑤))
2928biimpd 144 . . . . . . . . . 10 (𝑤 = ∅ → (suc 𝑦𝑤 → suc 𝑦 = 𝑤))
3029a1i 9 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑤 = ∅ → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
31 nfv 1528 . . . . . . . . . . 11 𝑧 𝑦 ∈ ω
32 nfra1 2508 . . . . . . . . . . 11 𝑧𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)
3331, 32nfan 1565 . . . . . . . . . 10 𝑧(𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧))
34 nfv 1528 . . . . . . . . . 10 𝑧(suc 𝑦𝑤 → suc 𝑦 = 𝑤)
35 rsp 2524 . . . . . . . . . . . . . 14 (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → (𝑧 ∈ ω → (𝑦𝑧𝑦 = 𝑧)))
36 vex 2740 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
37 vex 2740 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
3836, 37phplem4 6849 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → (suc 𝑦 ≈ suc 𝑧𝑦𝑧))
3938imim1d 75 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ 𝑧 ∈ ω) → ((𝑦𝑧𝑦 = 𝑧) → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧)))
4039ex 115 . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . 12 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (𝑧 ∈ ω → (suc 𝑦 ≈ suc 𝑧𝑦 = 𝑧)))
44 suceq 4399 . . . . . . . . . . . 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 4004 . . . . . . . . . . . . 13 (𝑤 = suc 𝑧 → (suc 𝑦𝑤 ↔ suc 𝑦 ≈ suc 𝑧))
47 eqeq2 2187 . . . . . . . . . . . . 13 (𝑤 = suc 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = suc 𝑧))
4846, 47imbi12d 234 . . . . . . . . . . . 12 (𝑤 = suc 𝑧 → ((suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦 ≈ suc 𝑧 → suc 𝑦 = suc 𝑧)))
4948biimprcd 160 . . . . . . . . . . 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 2591 . . . . . . . . 9 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → (∃𝑧 ∈ ω 𝑤 = suc 𝑧 → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5230, 51jaod 717 . . . . . . . 8 ((𝑦 ∈ ω ∧ ∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧)) → ((𝑤 = ∅ ∨ ∃𝑧 ∈ ω 𝑤 = suc 𝑧) → (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
5352ex 115 . . . . . . 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 2556 . . . . 5 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ∀𝑤 ∈ ω (suc 𝑦𝑤 → suc 𝑦 = 𝑤)))
56 breq2 4004 . . . . . . 7 (𝑤 = 𝑧 → (suc 𝑦𝑤 ↔ suc 𝑦𝑧))
57 eqeq2 2187 . . . . . . 7 (𝑤 = 𝑧 → (suc 𝑦 = 𝑤 ↔ suc 𝑦 = 𝑧))
5856, 57imbi12d 234 . . . . . 6 (𝑤 = 𝑧 → ((suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
5958cbvralv 2703 . . . . 5 (∀𝑤 ∈ ω (suc 𝑦𝑤 → suc 𝑦 = 𝑤) ↔ ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧))
6055, 59syl6ib 161 . . . 4 (𝑦 ∈ ω → (∀𝑧 ∈ ω (𝑦𝑧𝑦 = 𝑧) → ∀𝑧 ∈ ω (suc 𝑦𝑧 → suc 𝑦 = 𝑧)))
614, 8, 12, 16, 22, 60finds 4596 . . 3 (𝐴 ∈ ω → ∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧))
62 breq2 4004 . . . . 5 (𝑧 = 𝐵 → (𝐴𝑧𝐴𝐵))
63 eqeq2 2187 . . . . 5 (𝑧 = 𝐵 → (𝐴 = 𝑧𝐴 = 𝐵))
6462, 63imbi12d 234 . . . 4 (𝑧 = 𝐵 → ((𝐴𝑧𝐴 = 𝑧) ↔ (𝐴𝐵𝐴 = 𝐵)))
6564rspcv 2837 . . 3 (𝐵 ∈ ω → (∀𝑧 ∈ ω (𝐴𝑧𝐴 = 𝑧) → (𝐴𝐵𝐴 = 𝐵)))
6661, 65mpan9 281 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
67 eqeng 6760 . . 3 (𝐴 ∈ ω → (𝐴 = 𝐵𝐴𝐵))
6867adantr 276 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = 𝐵𝐴𝐵))
6966, 68impbid 129 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵𝐴 = 𝐵))
Colors of variables: wff set class
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
  Copyright terms: Public domain W3C validator