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Theorem bj-peano4 9343
Description: Remove from peano4 4263 dependency on ax-setind 4220. 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 900 . . . . 5 ((A 𝜔 B 𝜔 suc A = suc B) → (A 𝜔 B 𝜔))
2 pm3.22 252 . . . . 5 ((A 𝜔 B 𝜔) → (B 𝜔 A 𝜔))
3 bj-nnen2lp 9342 . . . . 5 ((B 𝜔 A 𝜔) → ¬ (B A A B))
41, 2, 33syl 17 . . . 4 ((A 𝜔 B 𝜔 suc A = suc B) → ¬ (B A A B))
5 sucidg 4119 . . . . . . . . . . . 12 (B 𝜔 → B suc B)
6 eleq2 2098 . . . . . . . . . . . 12 (suc A = suc B → (B suc AB suc B))
75, 6syl5ibrcom 146 . . . . . . . . . . 11 (B 𝜔 → (suc A = suc BB suc A))
8 elsucg 4107 . . . . . . . . . . 11 (B 𝜔 → (B suc A ↔ (B A B = A)))
97, 8sylibd 138 . . . . . . . . . 10 (B 𝜔 → (suc A = suc B → (B A B = A)))
109imp 115 . . . . . . . . 9 ((B 𝜔 suc A = suc B) → (B A B = A))
11103adant1 921 . . . . . . . 8 ((A 𝜔 B 𝜔 suc A = suc B) → (B A B = A))
12 sucidg 4119 . . . . . . . . . . . 12 (A 𝜔 → A suc A)
13 eleq2 2098 . . . . . . . . . . . 12 (suc A = suc B → (A suc AA suc B))
1412, 13syl5ibcom 144 . . . . . . . . . . 11 (A 𝜔 → (suc A = suc BA suc B))
15 elsucg 4107 . . . . . . . . . . 11 (A 𝜔 → (A suc B ↔ (A B A = B)))
1614, 15sylibd 138 . . . . . . . . . 10 (A 𝜔 → (suc A = suc B → (A B A = B)))
1716imp 115 . . . . . . . . 9 ((A 𝜔 suc A = suc B) → (A B A = B))
18173adant2 922 . . . . . . . 8 ((A 𝜔 B 𝜔 suc A = suc B) → (A B A = B))
1911, 18jca 290 . . . . . . 7 ((A 𝜔 B 𝜔 suc A = suc B) → ((B A B = A) (A B A = B)))
20 eqcom 2039 . . . . . . . . 9 (B = AA = B)
2120orbi2i 678 . . . . . . . 8 ((B A B = A) ↔ (B A A = B))
2221anbi1i 431 . . . . . . 7 (((B A B = A) (A B A = B)) ↔ ((B A A = B) (A B A = B)))
2319, 22sylib 127 . . . . . 6 ((A 𝜔 B 𝜔 suc A = suc B) → ((B A A = B) (A B A = B)))
24 ordir 729 . . . . . 6 (((B A A B) A = B) ↔ ((B A A = B) (A B A = B)))
2523, 24sylibr 137 . . . . 5 ((A 𝜔 B 𝜔 suc A = suc B) → ((B A A B) A = B))
2625ord 642 . . . 4 ((A 𝜔 B 𝜔 suc A = suc B) → (¬ (B A A B) → A = B))
274, 26mpd 13 . . 3 ((A 𝜔 B 𝜔 suc A = suc B) → A = B)
28273expia 1105 . 2 ((A 𝜔 B 𝜔) → (suc A = suc BA = B))
29 suceq 4105 . 2 (A = B → suc A = suc B)
3028, 29impbid1 130 1 ((A 𝜔 B 𝜔) → (suc A = suc BA = B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628   w3a 884   = wceq 1242   wcel 1390  suc csuc 4068  𝜔com 4256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9202  ax-bdor 9205  ax-bdn 9206  ax-bdal 9207  ax-bdex 9208  ax-bdeq 9209  ax-bdel 9210  ax-bdsb 9211  ax-bdsep 9273  ax-infvn 9329
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9230  df-bj-ind 9316
This theorem is referenced by: (None)
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