Theorem suc11reg 7824

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 7818	. . . . 5 |- -. ( A e. B /\ B e. A )
2		ianor 488	. . . . 5 |- ( -. ( A e. B /\ B e. A ) <-> ( -. A e. B \/ -. B e. A ) )
3	1, 2	mpbi 208	. . . 4 |- ( -. A e. B \/ -. B e. A )
4		sucidg 4796	. . . . . . . . . . 11 |- ( A e. _V -> A e. suc A )
5		eleq2 2503	. . . . . . . . . . 11 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
6	4, 5	syl5ibcom 220	. . . . . . . . . 10 |- ( A e. _V -> ( suc A = suc B -> A e. suc B ) )
7		elsucg 4785	. . . . . . . . . 10 |- ( A e. _V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
8	6, 7	sylibd 214	. . . . . . . . 9 |- ( A e. _V -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
9	8	imp 429	. . . . . . . 8 |- ( ( A e. _V /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
10	9	ord 377	. . . . . . 7 |- ( ( A e. _V /\ suc A = suc B ) -> ( -. A e. B -> A = B ) )
11	10	ex 434	. . . . . 6 |- ( A e. _V -> ( suc A = suc B -> ( -. A e. B -> A = B ) ) )
12	11	com23 78	. . . . 5 |- ( A e. _V -> ( -. A e. B -> ( suc A = suc B -> A = B ) ) )
13		sucidg 4796	. . . . . . . . . . . 12 |- ( B e. _V -> B e. suc B )
14		eleq2 2503	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
15	13, 14	syl5ibrcom 222	. . . . . . . . . . 11 |- ( B e. _V -> ( suc A = suc B -> B e. suc A ) )
16		elsucg 4785	. . . . . . . . . . 11 |- ( B e. _V -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
17	15, 16	sylibd 214	. . . . . . . . . 10 |- ( B e. _V -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
18	17	imp 429	. . . . . . . . 9 |- ( ( B e. _V /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
19	18	ord 377	. . . . . . . 8 |- ( ( B e. _V /\ suc A = suc B ) -> ( -. B e. A -> B = A ) )
20		eqcom 2444	. . . . . . . 8 |- ( B = A <-> A = B )
21	19, 20	syl6ib 226	. . . . . . 7 |- ( ( B e. _V /\ suc A = suc B ) -> ( -. B e. A -> A = B ) )
22	21	ex 434	. . . . . 6 |- ( B e. _V -> ( suc A = suc B -> ( -. B e. A -> A = B ) ) )
23	22	com23 78	. . . . 5 |- ( B e. _V -> ( -. B e. A -> ( suc A = suc B -> A = B ) ) )
24	12, 23	jaao 509	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( ( -. A e. B \/ -. B e. A ) -> ( suc A = suc B -> A = B ) ) )
25	3, 24	mpi 17	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( suc A = suc B -> A = B ) )
26		sucexb 6419	. . . . 5 |- ( A e. _V <-> suc A e. _V )
27		sucexb 6419	. . . . . 6 |- ( B e. _V <-> suc B e. _V )
28	27	notbii 296	. . . . 5 |- ( -. B e. _V <-> -. suc B e. _V )
29		nelneq 2540	. . . . 5 |- ( ( suc A e. _V /\ -. suc B e. _V ) -> -. suc A = suc B )
30	26, 28, 29	syl2anb 479	. . . 4 |- ( ( A e. _V /\ -. B e. _V ) -> -. suc A = suc B )
31	30	pm2.21d 106	. . 3 |- ( ( A e. _V /\ -. B e. _V ) -> ( suc A = suc B -> A = B ) )
32		eqcom 2444	. . . 4 |- ( suc A = suc B <-> suc B = suc A )
33	26	notbii 296	. . . . . . 7 |- ( -. A e. _V <-> -. suc A e. _V )
34		nelneq 2540	. . . . . . 7 |- ( ( suc B e. _V /\ -. suc A e. _V ) -> -. suc B = suc A )
35	27, 33, 34	syl2anb 479	. . . . . 6 |- ( ( B e. _V /\ -. A e. _V ) -> -. suc B = suc A )
36	35	ancoms 453	. . . . 5 |- ( ( -. A e. _V /\ B e. _V ) -> -. suc B = suc A )
37	36	pm2.21d 106	. . . 4 |- ( ( -. A e. _V /\ B e. _V ) -> ( suc B = suc A -> A = B ) )
38	32, 37	syl5bi 217	. . 3 |- ( ( -. A e. _V /\ B e. _V ) -> ( suc A = suc B -> A = B ) )
39		sucprc 4793	. . . . 5 |- ( -. A e. _V -> suc A = A )
40		sucprc 4793	. . . . 5 |- ( -. B e. _V -> suc B = B )
41	39, 40	eqeqan12d 2457	. . . 4 |- ( ( -. A e. _V /\ -. B e. _V ) -> ( suc A = suc B <-> A = B ) )
42	41	biimpd 207	. . 3 |- ( ( -. A e. _V /\ -. B e. _V ) -> ( suc A = suc B -> A = B ) )
43	25, 31, 38, 42	4cases 940	. 2 |- ( suc A = suc B -> A = B )
44		suceq 4783	. 2 |- ( A = B -> suc A = suc B )
45	43, 44	impbii 188	1 |- ( suc A = suc B <-> A = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   _Vcvv 2971   suc csuc 4720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371  ax-reg 7806

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-eprel 4631  df-fr 4678  df-suc 4724

This theorem is referenced by:  rankxpsuc  8088  bnj551  31732