Theorem suc11g 4594

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 4591	. . . 4 |- -. ( B e. A /\ A e. B )
2		sucidg 4452	. . . . . . . . . . . 12 |- ( B e. W -> B e. suc B )
3		eleq2 2260	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
4	2, 3	syl5ibrcom 157	. . . . . . . . . . 11 |- ( B e. W -> ( suc A = suc B -> B e. suc A ) )
5		elsucg 4440	. . . . . . . . . . 11 |- ( B e. W -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
6	4, 5	sylibd 149	. . . . . . . . . 10 |- ( B e. W -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
7	6	imp 124	. . . . . . . . 9 |- ( ( B e. W /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
8	7	3adant1 1017	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
9		sucidg 4452	. . . . . . . . . . . 12 |- ( A e. V -> A e. suc A )
10		eleq2 2260	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
11	9, 10	syl5ibcom 155	. . . . . . . . . . 11 |- ( A e. V -> ( suc A = suc B -> A e. suc B ) )
12		elsucg 4440	. . . . . . . . . . 11 |- ( A e. V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
13	11, 12	sylibd 149	. . . . . . . . . 10 |- ( A e. V -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
14	13	imp 124	. . . . . . . . 9 |- ( ( A e. V /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
15	14	3adant2 1018	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
16	8, 15	jca 306	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A \/ B = A ) /\ ( A e. B \/ A = B ) ) )
17		eqcom 2198	. . . . . . . . 9 |- ( B = A <-> A = B )
18	17	orbi2i 763	. . . . . . . 8 |- ( ( B e. A \/ B = A ) <-> ( B e. A \/ A = B ) )
19	18	anbi1i 458	. . . . . . 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 122	. . . . . 6 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
21		ordir 818	. . . . . 6 |- ( ( ( B e. A /\ A e. B ) \/ A = B ) <-> ( ( B e. A \/ A = B ) /\ ( A e. B \/ A = B ) ) )
22	20, 21	sylibr 134	. . . . 5 |- ( ( A e. V /\ B e. W /\ suc A = suc B ) -> ( ( B e. A /\ A e. B ) \/ A = B ) )
23	22	ord 725	. . . 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 1207	. 2 |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B -> A = B ) )
26		suceq 4438	. 2 |- ( A = B -> suc A = suc B )
27	25, 26	impbid1 142	1 |- ( ( A e. V /\ B e. W ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   suc csuc 4401

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4574

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-dif 3159  df-un 3161  df-sn 3629  df-pr 3630  df-suc 4407

This theorem is referenced by:  suc11  4595  peano4  4634  frecsuclem  6473