Theorem bj-peano4 15447

Description: Remove from peano4 4629 dependency on ax-setind 4569. 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 996	. . . . 5 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( A e. om /\ B e. om ) )
2		pm3.22 265	. . . . 5 |- ( ( A e. om /\ B e. om ) -> ( B e. om /\ A e. om ) )
3		bj-nnen2lp 15446	. . . . 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 4447	. . . . . . . . . . . 12 |- ( B e. om -> B e. suc B )
6		eleq2 2257	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
7	5, 6	syl5ibrcom 157	. . . . . . . . . . 11 |- ( B e. om -> ( suc A = suc B -> B e. suc A ) )
8		elsucg 4435	. . . . . . . . . . 11 |- ( B e. om -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
9	7, 8	sylibd 149	. . . . . . . . . 10 |- ( B e. om -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
10	9	imp 124	. . . . . . . . 9 |- ( ( B e. om /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
11	10	3adant1 1017	. . . . . . . 8 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
12		sucidg 4447	. . . . . . . . . . . 12 |- ( A e. om -> A e. suc A )
13		eleq2 2257	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
14	12, 13	syl5ibcom 155	. . . . . . . . . . 11 |- ( A e. om -> ( suc A = suc B -> A e. suc B ) )
15		elsucg 4435	. . . . . . . . . . 11 |- ( A e. om -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
16	14, 15	sylibd 149	. . . . . . . . . 10 |- ( A e. om -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
17	16	imp 124	. . . . . . . . 9 |- ( ( A e. om /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
18	17	3adant2 1018	. . . . . . . 8 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
19	11, 18	jca 306	. . . . . . 7 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( ( B e. A \/ B = A ) /\ ( A e. B \/ A = B ) ) )
20		eqcom 2195	. . . . . . . . 9 |- ( B = A <-> A = B )
21	20	orbi2i 763	. . . . . . . 8 |- ( ( B e. A \/ B = A ) <-> ( B e. A \/ A = B ) )
22	21	anbi1i 458	. . . . . . 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 122	. . . . . 6 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
24		ordir 818	. . . . . 6 |- ( ( ( B e. A /\ A e. B ) \/ A = B ) <-> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
25	23, 24	sylibr 134	. . . . 5 |- ( ( A e. om /\ B e. om /\ suc A = suc B ) -> ( ( B e. A /\ A e. B ) \/ A = B ) )
26	25	ord 725	. . . 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 1207	. 2 |- ( ( A e. om /\ B e. om ) -> ( suc A = suc B -> A = B ) )
29		suceq 4433	. 2 |- ( A = B -> suc A = suc B )
30	28, 29	impbid1 142	1 |- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   suc csuc 4396   omcom 4622

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-nul 4155  ax-pr 4238  ax-un 4464  ax-bd0 15305  ax-bdor 15308  ax-bdn 15309  ax-bdal 15310  ax-bdex 15311  ax-bdeq 15312  ax-bdel 15313  ax-bdsb 15314  ax-bdsep 15376  ax-infvn 15433

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-pr 3625  df-uni 3836  df-int 3871  df-suc 4402  df-iom 4623  df-bdc 15333  df-bj-ind 15419

This theorem is referenced by: (None)