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Theorem bj-peano4 13183
Description: Remove from peano4 4511 dependency on ax-setind 4452. Therefore, it only requires core constructive axioms (albeit more of them). (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-peano4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))

Proof of Theorem bj-peano4
StepHypRef Expression
1 3simpa 978 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐴 ∈ ω ∧ 𝐵 ∈ ω))
2 pm3.22 263 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ ω ∧ 𝐴 ∈ ω))
3 bj-nnen2lp 13182 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ¬ (𝐵𝐴𝐴𝐵))
41, 2, 33syl 17 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ¬ (𝐵𝐴𝐴𝐵))
5 sucidg 4338 . . . . . . . . . . . 12 (𝐵 ∈ ω → 𝐵 ∈ suc 𝐵)
6 eleq2 2203 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐵 ∈ suc 𝐴𝐵 ∈ suc 𝐵))
75, 6syl5ibrcom 156 . . . . . . . . . . 11 (𝐵 ∈ ω → (suc 𝐴 = suc 𝐵𝐵 ∈ suc 𝐴))
8 elsucg 4326 . . . . . . . . . . 11 (𝐵 ∈ ω → (𝐵 ∈ suc 𝐴 ↔ (𝐵𝐴𝐵 = 𝐴)))
97, 8sylibd 148 . . . . . . . . . 10 (𝐵 ∈ ω → (suc 𝐴 = suc 𝐵 → (𝐵𝐴𝐵 = 𝐴)))
109imp 123 . . . . . . . . 9 ((𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
11103adant1 999 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐵𝐴𝐵 = 𝐴))
12 sucidg 4338 . . . . . . . . . . . 12 (𝐴 ∈ ω → 𝐴 ∈ suc 𝐴)
13 eleq2 2203 . . . . . . . . . . . 12 (suc 𝐴 = suc 𝐵 → (𝐴 ∈ suc 𝐴𝐴 ∈ suc 𝐵))
1412, 13syl5ibcom 154 . . . . . . . . . . 11 (𝐴 ∈ ω → (suc 𝐴 = suc 𝐵𝐴 ∈ suc 𝐵))
15 elsucg 4326 . . . . . . . . . . 11 (𝐴 ∈ ω → (𝐴 ∈ suc 𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
1614, 15sylibd 148 . . . . . . . . . 10 (𝐴 ∈ ω → (suc 𝐴 = suc 𝐵 → (𝐴𝐵𝐴 = 𝐵)))
1716imp 123 . . . . . . . . 9 ((𝐴 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
18173adant2 1000 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (𝐴𝐵𝐴 = 𝐵))
1911, 18jca 304 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)))
20 eqcom 2141 . . . . . . . . 9 (𝐵 = 𝐴𝐴 = 𝐵)
2120orbi2i 751 . . . . . . . 8 ((𝐵𝐴𝐵 = 𝐴) ↔ (𝐵𝐴𝐴 = 𝐵))
2221anbi1i 453 . . . . . . 7 (((𝐵𝐴𝐵 = 𝐴) ∧ (𝐴𝐵𝐴 = 𝐵)) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2319, 22sylib 121 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
24 ordir 806 . . . . . 6 (((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵) ↔ ((𝐵𝐴𝐴 = 𝐵) ∧ (𝐴𝐵𝐴 = 𝐵)))
2523, 24sylibr 133 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → ((𝐵𝐴𝐴𝐵) ∨ 𝐴 = 𝐵))
2625ord 713 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → (¬ (𝐵𝐴𝐴𝐵) → 𝐴 = 𝐵))
274, 26mpd 13 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ suc 𝐴 = suc 𝐵) → 𝐴 = 𝐵)
28273expia 1183 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
29 suceq 4324 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
3028, 29impbid1 141 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 = suc 𝐵𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  w3a 962   = wceq 1331  wcel 1480  suc csuc 4287  ωcom 4504
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13041  ax-bdor 13044  ax-bdn 13045  ax-bdal 13046  ax-bdex 13047  ax-bdeq 13048  ax-bdel 13049  ax-bdsb 13050  ax-bdsep 13112  ax-infvn 13169
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13069  df-bj-ind 13155
This theorem is referenced by: (None)
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