Theorem suc11g 4534

Description: The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.)

Assertion

Ref	Expression
suc11g	|- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )

Proof of Theorem suc11g

Step	Hyp	Ref	Expression
1		en2lp 4531	. . . 4 |- -. ( B e. A /\ A e. B )
2		sucidg 4394	. . . . . . . . . . . 12 |- ( B e. W -> B e. suc B )
3		eleq2 2230	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
4	2, 3	syl5ibrcom 156	. . . . . . . . . . 11 |- ( B e. W -> ( suc A = suc B -> B e. suc A ) )
5		elsucg 4382	. . . . . . . . . . 11 |- ( B e. W -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
6	4, 5	sylibd 148	. . . . . . . . . 10 |- ( B e. W -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
7	6	imp 123	. . . . . . . . 9 |- ( ( B e. W /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
8	7	3adant1 1005	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
9		sucidg 4394	. . . . . . . . . . . 12 |- ( A e. V -> A e. suc A )
10		eleq2 2230	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
11	9, 10	syl5ibcom 154	. . . . . . . . . . 11 |- ( A e. V -> ( suc A = suc B -> A e. suc B ) )
12		elsucg 4382	. . . . . . . . . . 11 |- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
13	11, 12	sylibd 148	. . . . . . . . . 10 |- ( A e. V -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
14	13	imp 123	. . . . . . . . 9 |- ( ( A e. V /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
15	14	3adant2 1006	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
16	8, 15	jca 304	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A \/ B = A ) /\ ( A e. B \/ A = B ) ) )
17		eqcom 2167	. . . . . . . . 9 |- ( B = A <-> A = B )
18	17	orbi2i 752	. . . . . . . 8 |- ( ( B e. A \/ B = A ) <-> ( B e. A \/ A = B ) )
19	18	anbi1i 454	. . . . . . 7 |- ( ( ( B e. A \/ B = A ) /\ ( A e. B \/ A = B ) ) <-> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
20	16, 19	sylib 121	. . . . . 6 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
21		ordir 807	. . . . . 6 |- ( ( ( B e. A /\ A e. B ) \/ A = B ) <-> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
22	20, 21	sylibr 133	. . . . 5 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A /\ A e. B ) \/ A = B ) )
23	22	ord 714	. . . 4 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( -. ( B e. A /\ A e. B ) -> A = B ) )
24	1, 23	mpi 15	. . 3 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> A = B )
25	24	3expia 1195	. 2 |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B -> A = B ) )
26		suceq 4380	. 2 |- ( A = B -> suc A = suc B )
27	25, 26	impbid1 141	1 |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   suc csuc 4343

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-dif 3118  df-un 3120  df-sn 3582  df-pr 3583  df-suc 4349

This theorem is referenced by:  suc11  4535  peano4  4574  frecsuclem  6374