Theorem bj-peano4 10750

Description: Remove from peano4 4338 dependency on ax-setind 4280. 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

Step	Hyp	Ref	Expression
1		3simpa 935	. . . . 5 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( A e. om /\ B e. om ) )
2		pm3.22 261	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( B e. om /\ A e. om ) )
3		bj-nnen2lp 10749	. . . . 5 |- ( ( B e. om /\ A e. om ) -> -. ( B e. A /\ A e. B ) )
4	1, 2, 3	3syl 17	. . . 4 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> -. ( B e. A /\ A e. B ) )
5		sucidg 4171	. . . . . . . . . . . 12 |- ( B e. om -> B e. suc B )
6		eleq2 2142	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
7	5, 6	syl5ibrcom 155	. . . . . . . . . . 11 |- ( B e. om -> ( suc A = suc B -> B e. suc A ) )
8		elsucg 4159	. . . . . . . . . . 11 |- ( B e. om -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
9	7, 8	sylibd 147	. . . . . . . . . 10 |- ( B e. om -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
10	9	imp 122	. . . . . . . . 9 |- ( ( B e. om /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
11	10	3adant1 956	. . . . . . . 8 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
12		sucidg 4171	. . . . . . . . . . . 12 |- ( A e. om -> A e. suc A )
13		eleq2 2142	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
14	12, 13	syl5ibcom 153	. . . . . . . . . . 11 |- ( A e. om -> ( suc A = suc B -> A e. suc B ) )
15		elsucg 4159	. . . . . . . . . . 11 |- ( A e. om -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
16	14, 15	sylibd 147	. . . . . . . . . 10 |- ( A e. om -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
17	16	imp 122	. . . . . . . . 9 |- ( ( A e. om /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
18	17	3adant2 957	. . . . . . . 8 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
19	11, 18	jca 300	. . . . . . 7 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( ( B e. A \/ B = A ) /\ ( A e. B \/ A = B ) ) )
20		eqcom 2083	. . . . . . . . 9 |- ( B = A <-> A = B )
21	20	orbi2i 711	. . . . . . . 8 |- ( ( B e. A \/ B = A ) <-> ( B e. A \/ A = B ) )
22	21	anbi1i 445	. . . . . . 7 |- ( ( ( B e. A \/ B = A ) /\ ( A e. B \/ A = B ) ) <-> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
23	19, 22	sylib 120	. . . . . 6 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
24		ordir 763	. . . . . 6 |- ( ( ( B e. A /\ A e. B ) \/ A = B ) <-> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
25	23, 24	sylibr 132	. . . . 5 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( ( B e. A /\ A e. B ) \/ A = B ) )
26	25	ord 675	. . . 4 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( -. ( B e. A /\ A e. B ) -> A = B ) )
27	4, 26	mpd 13	. . 3 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> A = B )
28	27	3expia 1140	. 2 |- ( ( A e. om /\ B e. om ) -> ( suc A = suc B -> A = B ) )
29		suceq 4157	. 2 |- ( A = B -> suc A = suc B )
30	28, 29	impbid1 140	1 |- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   /\ w3a 919   = wceq 1284   e. wcel 1433   suc csuc 4120   omcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-nul 3904  ax-pr 3964  ax-un 4188  ax-bd0 10604  ax-bdor 10607  ax-bdn 10608  ax-bdal 10609  ax-bdex 10610  ax-bdeq 10611  ax-bdel 10612  ax-bdsb 10613  ax-bdsep 10675  ax-infvn 10736

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332  df-bdc 10632  df-bj-ind 10722

This theorem is referenced by: (None)