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Theorem bj-peano4 10467
Description: Remove from peano4 4348 dependency on ax-setind 4290. 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  e.  om  /\  B  e.  om )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )

Proof of Theorem bj-peano4
StepHypRef Expression
1 3simpa 912 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om  /\  suc  A  =  suc  B )  ->  ( A  e. 
om  /\  B  e.  om ) )
2 pm3.22 256 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( B  e.  om  /\  A  e.  om )
)
3 bj-nnen2lp 10466 . . . . 5  |-  ( ( B  e.  om  /\  A  e.  om )  ->  -.  ( B  e.  A  /\  A  e.  B ) )
41, 2, 33syl 17 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om  /\  suc  A  =  suc  B )  ->  -.  ( B  e.  A  /\  A  e.  B ) )
5 sucidg 4181 . . . . . . . . . . . 12  |-  ( B  e.  om  ->  B  e.  suc  B )
6 eleq2 2117 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( B  e.  suc  A  <-> 
B  e.  suc  B
) )
75, 6syl5ibrcom 150 . . . . . . . . . . 11  |-  ( B  e.  om  ->  ( suc  A  =  suc  B  ->  B  e.  suc  A
) )
8 elsucg 4169 . . . . . . . . . . 11  |-  ( B  e.  om  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
97, 8sylibd 142 . . . . . . . . . 10  |-  ( B  e.  om  ->  ( suc  A  =  suc  B  ->  ( B  e.  A  \/  B  =  A
) ) )
109imp 119 . . . . . . . . 9  |-  ( ( B  e.  om  /\  suc  A  =  suc  B
)  ->  ( B  e.  A  \/  B  =  A ) )
11103adant1 933 . . . . . . . 8  |-  ( ( A  e.  om  /\  B  e.  om  /\  suc  A  =  suc  B )  ->  ( B  e.  A  \/  B  =  A ) )
12 sucidg 4181 . . . . . . . . . . . 12  |-  ( A  e.  om  ->  A  e.  suc  A )
13 eleq2 2117 . . . . . . . . . . . 12  |-  ( suc 
A  =  suc  B  ->  ( A  e.  suc  A  <-> 
A  e.  suc  B
) )
1412, 13syl5ibcom 148 . . . . . . . . . . 11  |-  ( A  e.  om  ->  ( suc  A  =  suc  B  ->  A  e.  suc  B
) )
15 elsucg 4169 . . . . . . . . . . 11  |-  ( A  e.  om  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
1614, 15sylibd 142 . . . . . . . . . 10  |-  ( A  e.  om  ->  ( suc  A  =  suc  B  ->  ( A  e.  B  \/  A  =  B
) ) )
1716imp 119 . . . . . . . . 9  |-  ( ( A  e.  om  /\  suc  A  =  suc  B
)  ->  ( A  e.  B  \/  A  =  B ) )
18173adant2 934 . . . . . . . 8  |-  ( ( A  e.  om  /\  B  e.  om  /\  suc  A  =  suc  B )  ->  ( A  e.  B  \/  A  =  B ) )
1911, 18jca 294 . . . . . . 7  |-  ( ( A  e.  om  /\  B  e.  om  /\  suc  A  =  suc  B )  ->  ( ( B  e.  A  \/  B  =  A )  /\  ( A  e.  B  \/  A  =  B )
) )
20 eqcom 2058 . . . . . . . . 9  |-  ( B  =  A  <->  A  =  B )
2120orbi2i 689 . . . . . . . 8  |-  ( ( B  e.  A  \/  B  =  A )  <->  ( B  e.  A  \/  A  =  B )
)
2221anbi1i 439 . . . . . . 7  |-  ( ( ( B  e.  A  \/  B  =  A
)  /\  ( A  e.  B  \/  A  =  B ) )  <->  ( ( B  e.  A  \/  A  =  B )  /\  ( A  e.  B  \/  A  =  B
) ) )
2319, 22sylib 131 . . . . . 6  |-  ( ( A  e.  om  /\  B  e.  om  /\  suc  A  =  suc  B )  ->  ( ( B  e.  A  \/  A  =  B )  /\  ( A  e.  B  \/  A  =  B )
) )
24 ordir 741 . . . . . 6  |-  ( ( ( B  e.  A  /\  A  e.  B
)  \/  A  =  B )  <->  ( ( B  e.  A  \/  A  =  B )  /\  ( A  e.  B  \/  A  =  B
) ) )
2523, 24sylibr 141 . . . . 5  |-  ( ( A  e.  om  /\  B  e.  om  /\  suc  A  =  suc  B )  ->  ( ( B  e.  A  /\  A  e.  B )  \/  A  =  B ) )
2625ord 653 . . . 4  |-  ( ( A  e.  om  /\  B  e.  om  /\  suc  A  =  suc  B )  ->  ( -.  ( B  e.  A  /\  A  e.  B )  ->  A  =  B ) )
274, 26mpd 13 . . 3  |-  ( ( A  e.  om  /\  B  e.  om  /\  suc  A  =  suc  B )  ->  A  =  B )
28273expia 1117 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  =  suc  B  ->  A  =  B ) )
29 suceq 4167 . 2  |-  ( A  =  B  ->  suc  A  =  suc  B )
3028, 29impbid1 134 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   suc csuc 4130   omcom 4341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3911  ax-pr 3972  ax-un 4198  ax-bd0 10320  ax-bdor 10323  ax-bdn 10324  ax-bdal 10325  ax-bdex 10326  ax-bdeq 10327  ax-bdel 10328  ax-bdsb 10329  ax-bdsep 10391  ax-infvn 10453
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-suc 4136  df-iom 4342  df-bdc 10348  df-bj-ind 10438
This theorem is referenced by: (None)
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