Theorem suc11reg 8155

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 8149	. . . . 5 |- -. ( A e. B /\ B e. A )
2		ianor 495	. . . . 5 |- ( -. ( A e. B /\ B e. A ) <-> ( -. A e. B \/ -. B e. A ) )
3	1, 2	mpbi 213	. . . 4 |- ( -. A e. B \/ -. B e. A )
4		sucidg 5524	. . . . . . . . . . 11 |- ( A e. _V -> A e. suc A )
5		eleq2 2529	. . . . . . . . . . 11 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
6	4, 5	syl5ibcom 228	. . . . . . . . . 10 |- ( A e. _V -> ( suc A = suc B -> A e. suc B ) )
7		elsucg 5513	. . . . . . . . . 10 |- ( A e. _V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
8	6, 7	sylibd 222	. . . . . . . . 9 |- ( A e. _V -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
9	8	imp 435	. . . . . . . 8 |- ( ( A e. _V /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
10	9	ord 383	. . . . . . 7 |- ( ( A e. _V /\ suc A = suc B ) -> ( -. A e. B -> A = B ) )
11	10	ex 440	. . . . . 6 |- ( A e. _V -> ( suc A = suc B -> ( -. A e. B -> A = B ) ) )
12	11	com23 81	. . . . 5 |- ( A e. _V -> ( -. A e. B -> ( suc A = suc B -> A = B ) ) )
13		sucidg 5524	. . . . . . . . . . . 12 |- ( B e. _V -> B e. suc B )
14		eleq2 2529	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
15	13, 14	syl5ibrcom 230	. . . . . . . . . . 11 |- ( B e. _V -> ( suc A = suc B -> B e. suc A ) )
16		elsucg 5513	. . . . . . . . . . 11 |- ( B e. _V -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
17	15, 16	sylibd 222	. . . . . . . . . 10 |- ( B e. _V -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
18	17	imp 435	. . . . . . . . 9 |- ( ( B e. _V /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
19	18	ord 383	. . . . . . . 8 |- ( ( B e. _V /\ suc A = suc B ) -> ( -. B e. A -> B = A ) )
20		eqcom 2469	. . . . . . . 8 |- ( B = A <-> A = B )
21	19, 20	syl6ib 234	. . . . . . 7 |- ( ( B e. _V /\ suc A = suc B ) -> ( -. B e. A -> A = B ) )
22	21	ex 440	. . . . . 6 |- ( B e. _V -> ( suc A = suc B -> ( -. B e. A -> A = B ) ) )
23	22	com23 81	. . . . 5 |- ( B e. _V -> ( -. B e. A -> ( suc A = suc B -> A = B ) ) )
24	12, 23	jaao 516	. . . 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 6668	. . . . 5 |- ( A e. _V <-> suc A e. _V )
27		sucexb 6668	. . . . . 6 |- ( B e. _V <-> suc B e. _V )
28	27	notbii 302	. . . . 5 |- ( -. B e. _V <-> -. suc B e. _V )
29		nelneq 2564	. . . . 5 |- ( ( suc A e. _V /\ -. suc B e. _V ) -> -. suc A = suc B )
30	26, 28, 29	syl2anb 486	. . . 4 |- ( ( A e. _V /\ -. B e. _V ) -> -. suc A = suc B )
31	30	pm2.21d 110	. . 3 |- ( ( A e. _V /\ -. B e. _V ) -> ( suc A = suc B -> A = B ) )
32		eqcom 2469	. . . 4 |- ( suc A = suc B <-> suc B = suc A )
33	26	notbii 302	. . . . . . 7 |- ( -. A e. _V <-> -. suc A e. _V )
34		nelneq 2564	. . . . . . 7 |- ( ( suc B e. _V /\ -. suc A e. _V ) -> -. suc B = suc A )
35	27, 33, 34	syl2anb 486	. . . . . 6 |- ( ( B e. _V /\ -. A e. _V ) -> -. suc B = suc A )
36	35	ancoms 459	. . . . 5 |- ( ( -. A e. _V /\ B e. _V ) -> -. suc B = suc A )
37	36	pm2.21d 110	. . . 4 |- ( ( -. A e. _V /\ B e. _V ) -> ( suc B = suc A -> A = B ) )
38	32, 37	syl5bi 225	. . 3 |- ( ( -. A e. _V /\ B e. _V ) -> ( suc A = suc B -> A = B ) )
39		sucprc 5521	. . . . 5 |- ( -. A e. _V -> suc A = A )
40		sucprc 5521	. . . . 5 |- ( -. B e. _V -> suc B = B )
41	39, 40	eqeqan12d 2478	. . . 4 |- ( ( -. A e. _V /\ -. B e. _V ) -> ( suc A = suc B <-> A = B ) )
42	41	biimpd 212	. . 3 |- ( ( -. A e. _V /\ -. B e. _V ) -> ( suc A = suc B -> A = B ) )
43	25, 31, 38, 42	4cases 966	. 2 |- ( suc A = suc B -> A = B )
44		suceq 5511	. 2 |- ( A = B -> suc A = suc B )
45	43, 44	impbii 192	1 |- ( suc A = suc B <-> A = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1455   e. wcel 1898   _Vcvv 3057   suc csuc 5448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656  ax-un 6615  ax-reg 8138

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-eprel 4767  df-fr 4815  df-suc 5452

This theorem is referenced by:  rankxpsuc  8384  bnj551  29602  1oequni2o  31817