Theorem suc11reg 7530

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 7527	. . . . 5 |- -. ( A e. B /\ B e. A )
2		ianor 475	. . . . 5 |- ( -. ( A e. B /\ B e. A ) <-> ( -. A e. B \/ -. B e. A ) )
3	1, 2	mpbi 200	. . . 4 |- ( -. A e. B \/ -. B e. A )
4		sucidg 4619	. . . . . . . . . . 11 |- ( A e. _V -> A e. suc A )
5		eleq2 2465	. . . . . . . . . . 11 |- ( suc A = suc B -> ( A e. suc A <-> A e. suc B ) )
6	4, 5	syl5ibcom 212	. . . . . . . . . 10 |- ( A e. _V -> ( suc A = suc B -> A e. suc B ) )
7		elsucg 4608	. . . . . . . . . 10 |- ( A e. _V -> ( A e. suc B <-> ( A e. B \/ A = B ) ) )
8	6, 7	sylibd 206	. . . . . . . . 9 |- ( A e. _V -> ( suc A = suc B -> ( A e. B \/ A = B ) ) )
9	8	imp 419	. . . . . . . 8 |- ( ( A e. _V /\ suc A = suc B ) -> ( A e. B \/ A = B ) )
10	9	ord 367	. . . . . . 7 |- ( ( A e. _V /\ suc A = suc B ) -> ( -. A e. B -> A = B ) )
11	10	ex 424	. . . . . 6 |- ( A e. _V -> ( suc A = suc B -> ( -. A e. B -> A = B ) ) )
12	11	com23 74	. . . . 5 |- ( A e. _V -> ( -. A e. B -> ( suc A = suc B -> A = B ) ) )
13		sucidg 4619	. . . . . . . . . . . 12 |- ( B e. _V -> B e. suc B )
14		eleq2 2465	. . . . . . . . . . . 12 |- ( suc A = suc B -> ( B e. suc A <-> B e. suc B ) )
15	13, 14	syl5ibrcom 214	. . . . . . . . . . 11 |- ( B e. _V -> ( suc A = suc B -> B e. suc A ) )
16		elsucg 4608	. . . . . . . . . . 11 |- ( B e. _V -> ( B e. suc A <-> ( B e. A \/ B = A ) ) )
17	15, 16	sylibd 206	. . . . . . . . . 10 |- ( B e. _V -> ( suc A = suc B -> ( B e. A \/ B = A ) ) )
18	17	imp 419	. . . . . . . . 9 |- ( ( B e. _V /\ suc A = suc B ) -> ( B e. A \/ B = A ) )
19	18	ord 367	. . . . . . . 8 |- ( ( B e. _V /\ suc A = suc B ) -> ( -. B e. A -> B = A ) )
20		eqcom 2406	. . . . . . . 8 |- ( B = A <-> A = B )
21	19, 20	syl6ib 218	. . . . . . 7 |- ( ( B e. _V /\ suc A = suc B ) -> ( -. B e. A -> A = B ) )
22	21	ex 424	. . . . . 6 |- ( B e. _V -> ( suc A = suc B -> ( -. B e. A -> A = B ) ) )
23	22	com23 74	. . . . 5 |- ( B e. _V -> ( -. B e. A -> ( suc A = suc B -> A = B ) ) )
24	12, 23	jaao 496	. . . 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 4748	. . . . 5 |- ( A e. _V <-> suc A e. _V )
27		sucexb 4748	. . . . . 6 |- ( B e. _V <-> suc B e. _V )
28	27	notbii 288	. . . . 5 |- ( -. B e. _V <-> -. suc B e. _V )
29		nelneq 2502	. . . . 5 |- ( ( suc A e. _V /\ -. suc B e. _V ) -> -. suc A = suc B )
30	26, 28, 29	syl2anb 466	. . . 4 |- ( ( A e. _V /\ -. B e. _V ) -> -. suc A = suc B )
31	30	pm2.21d 100	. . 3 |- ( ( A e. _V /\ -. B e. _V ) -> ( suc A = suc B -> A = B ) )
32		eqcom 2406	. . . 4 |- ( suc A = suc B <-> suc B = suc A )
33	26	notbii 288	. . . . . . 7 |- ( -. A e. _V <-> -. suc A e. _V )
34		nelneq 2502	. . . . . . 7 |- ( ( suc B e. _V /\ -. suc A e. _V ) -> -. suc B = suc A )
35	27, 33, 34	syl2anb 466	. . . . . 6 |- ( ( B e. _V /\ -. A e. _V ) -> -. suc B = suc A )
36	35	ancoms 440	. . . . 5 |- ( ( -. A e. _V /\ B e. _V ) -> -. suc B = suc A )
37	36	pm2.21d 100	. . . 4 |- ( ( -. A e. _V /\ B e. _V ) -> ( suc B = suc A -> A = B ) )
38	32, 37	syl5bi 209	. . 3 |- ( ( -. A e. _V /\ B e. _V ) -> ( suc A = suc B -> A = B ) )
39		sucprc 4616	. . . . 5 |- ( -. A e. _V -> suc A = A )
40		sucprc 4616	. . . . 5 |- ( -. B e. _V -> suc B = B )
41	39, 40	eqeqan12d 2419	. . . 4 |- ( ( -. A e. _V /\ -. B e. _V ) -> ( suc A = suc B <-> A = B ) )
42	41	biimpd 199	. . 3 |- ( ( -. A e. _V /\ -. B e. _V ) -> ( suc A = suc B -> A = B ) )
43	25, 31, 38, 42	4cases 916	. 2 |- ( suc A = suc B -> A = B )
44		suceq 4606	. 2 |- ( A = B -> suc A = suc B )
45	43, 44	impbii 181	1 |- ( suc A = suc B <-> A = B )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2916   suc csuc 4543

This theorem is referenced by:  rankxpsuc  7762  bnj551  28816

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660  ax-reg 7516

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-eprel 4454  df-fr 4501  df-suc 4547