Theorem suc11g 4281

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 4278	. . . 4 |- -. ( B e. A /\ A e. B )
2		sucidg 4153	. . . . . . . . . . . 12 |- ( B e. W -> B e. suc B )
3		eleq2 2101	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
4	2, 3	syl5ibrcom 146	. . . . . . . . . . 11 |- ( B e. W -> ( suc A = suc B -> B e. suc A ) )
5		elsucg 4141	. . . . . . . . . . 11 |- ( B e. W -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
6	4, 5	sylibd 138	. . . . . . . . . 10 |- ( B e. W -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
7	6	imp 115	. . . . . . . . 9 |- ( ( B e. W /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
8	7	3adant1 922	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
9		sucidg 4153	. . . . . . . . . . . 12 |- ( A e. V -> A e. suc A )
10		eleq2 2101	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
11	9, 10	syl5ibcom 144	. . . . . . . . . . 11 |- ( A e. V -> ( suc A = suc B -> A e. suc B ) )
12		elsucg 4141	. . . . . . . . . . 11 |- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
13	11, 12	sylibd 138	. . . . . . . . . 10 |- ( A e. V -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
14	13	imp 115	. . . . . . . . 9 |- ( ( A e. V /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
15	14	3adant2 923	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
16	8, 15	jca 290	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A \/ B = A ) /\ ( A e. B \/ A = B ) ) )
17		eqcom 2042	. . . . . . . . 9 |- ( B = A <-> A = B )
18	17	orbi2i 679	. . . . . . . 8 |- ( ( B e. A \/ B = A ) <-> ( B e. A \/ A = B ) )
19	18	anbi1i 431	. . . . . . 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 127	. . . . . 6 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
21		ordir 730	. . . . . 6 |- ( ( ( B e. A /\ A e. B ) \/ A = B ) <-> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
22	20, 21	sylibr 137	. . . . 5 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A /\ A e. B ) \/ A = B ) )
23	22	ord 643	. . . 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 1106	. 2 |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B -> A = B ) )
26		suceq 4139	. 2 |- ( A = B -> suc A = suc B )
27	25, 26	impbid1 130	1 |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   /\ w3a 885   = wceq 1243   e. wcel 1393   suc csuc 4102

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-sn 3381  df-pr 3382  df-suc 4108

This theorem is referenced by:  suc11  4282  peano4  4320  frecsuclem3  5990