Theorem suc11reg 8516

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

Assertion

Ref	Expression
suc11reg	|- ( suc A = suc B <-> A = B )

Proof of Theorem suc11reg

Step	Hyp	Ref	Expression
1		en2lp 8510	. . . . 5 |- -. ( A e. B /\ B e. A )
2		ianor 509	. . . . 5 |- ( -. ( A e. B /\ B e. A ) <-> ( -. A e. B \/ -. B e. A ) )
3	1, 2	mpbi 220	. . . 4 |- ( -. A e. B \/ -. B e. A )
4		sucidg 5803	. . . . . . . . . . 11 |- ( A e. _V -> A e. suc A )
5		eleq2 2690	. . . . . . . . . . 11 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
6	4, 5	syl5ibcom 235	. . . . . . . . . 10 |- ( A e. _V -> ( suc A = suc B -> A e. suc B ) )
7		elsucg 5792	. . . . . . . . . 10 |- ( A e. _V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
8	6, 7	sylibd 229	. . . . . . . . 9 |- ( A e. _V -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
9	8	imp 445	. . . . . . . 8 |- ( ( A e. _V /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
10	9	ord 392	. . . . . . 7 |- ( ( A e. _V /\ suc A = suc B ) -> ( -. A e. B -> A = B ) )
11	10	ex 450	. . . . . 6 |- ( A e. _V -> ( suc A = suc B -> ( -. A e. B -> A = B ) ) )
12	11	com23 86	. . . . 5 |- ( A e. _V -> ( -. A e. B -> ( suc A = suc B -> A = B ) ) )
13		sucidg 5803	. . . . . . . . . . . 12 |- ( B e. _V -> B e. suc B )
14		eleq2 2690	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
15	13, 14	syl5ibrcom 237	. . . . . . . . . . 11 |- ( B e. _V -> ( suc A = suc B -> B e. suc A ) )
16		elsucg 5792	. . . . . . . . . . 11 |- ( B e. _V -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
17	15, 16	sylibd 229	. . . . . . . . . 10 |- ( B e. _V -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
18	17	imp 445	. . . . . . . . 9 |- ( ( B e. _V /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
19	18	ord 392	. . . . . . . 8 |- ( ( B e. _V /\ suc A = suc B ) -> ( -. B e. A -> B = A ) )
20		eqcom 2629	. . . . . . . 8 |- ( B = A <-> A = B )
21	19, 20	syl6ib 241	. . . . . . 7 |- ( ( B e. _V /\ suc A = suc B ) -> ( -. B e. A -> A = B ) )
22	21	ex 450	. . . . . 6 |- ( B e. _V -> ( suc A = suc B -> ( -. B e. A -> A = B ) ) )
23	22	com23 86	. . . . 5 |- ( B e. _V -> ( -. B e. A -> ( suc A = suc B -> A = B ) ) )
24	12, 23	jaao 531	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( ( -. A e. B \/ -. B e. A ) -> ( suc A = suc B -> A = B ) ) )
25	3, 24	mpi 20	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( suc A = suc B -> A = B ) )
26		sucexb 7009	. . . . 5 |- ( A e. _V <-> suc A e. _V )
27		sucexb 7009	. . . . . 6 |- ( B e. _V <-> suc B e. _V )
28	27	notbii 310	. . . . 5 |- ( -. B e. _V <-> -. suc B e. _V )
29		nelneq 2725	. . . . 5 |- ( ( suc A e. _V /\ -. suc B e. _V ) -> -. suc A = suc B )
30	26, 28, 29	syl2anb 496	. . . 4 |- ( ( A e. _V /\ -. B e. _V ) -> -. suc A = suc B )
31	30	pm2.21d 118	. . 3 |- ( ( A e. _V /\ -. B e. _V ) -> ( suc A = suc B -> A = B ) )
32		eqcom 2629	. . . 4 |- ( suc A = suc B <-> suc B = suc A )
33	26	notbii 310	. . . . . . 7 |- ( -. A e. _V <-> -. suc A e. _V )
34		nelneq 2725	. . . . . . 7 |- ( ( suc B e. _V /\ -. suc A e. _V ) -> -. suc B = suc A )
35	27, 33, 34	syl2anb 496	. . . . . 6 |- ( ( B e. _V /\ -. A e. _V ) -> -. suc B = suc A )
36	35	ancoms 469	. . . . 5 |- ( ( -. A e. _V /\ B e. _V ) -> -. suc B = suc A )
37	36	pm2.21d 118	. . . 4 |- ( ( -. A e. _V /\ B e. _V ) -> ( suc B = suc A -> A = B ) )
38	32, 37	syl5bi 232	. . 3 |- ( ( -. A e. _V /\ B e. _V ) -> ( suc A = suc B -> A = B ) )
39		sucprc 5800	. . . . 5 |- ( -. A e. _V -> suc A = A )
40		sucprc 5800	. . . . 5 |- ( -. B e. _V -> suc B = B )
41	39, 40	eqeqan12d 2638	. . . 4 |- ( ( -. A e. _V /\ -. B e. _V ) -> ( suc A = suc B <-> A = B ) )
42	41	biimpd 219	. . 3 |- ( ( -. A e. _V /\ -. B e. _V ) -> ( suc A = suc B -> A = B ) )
43	25, 31, 38, 42	4cases 990	. 2 |- ( suc A = suc B -> A = B )
44		suceq 5790	. 2 |- ( A = B -> suc A = suc B )
45	43, 44	impbii 199	1 |- ( suc A = suc B <-> A = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073  df-suc 5729

This theorem is referenced by:  rankxpsuc  8745  bnj551  30812  1oequni2o  33216  clsk1indlem1  38343