Theorem suc11 5504

Description: The successor operation behaves like a one-to-one function. Compare Exercise 16 of [Enderton] p. 194. (Contributed by NM, 3-Sep-2003.)

Assertion

Ref	Expression
suc11	|- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) )

Proof of Theorem suc11

Step	Hyp	Ref	Expression
1		eloni 5411	. . . . 5 |- ( A e. On -> Ord A )
2		ordn2lp 5421	. . . . . 6 |- ( Ord A -> -. ( A e. B /\ B e. A ) )
3		ianor 495	. . . . . 6 |- ( -. ( A e. B /\ B e. A ) <-> ( -. A e. B \/ -. B e. A ) )
4	2, 3	sylib 201	. . . . 5 |- ( Ord A -> ( -. A e. B \/ -. B e. A ) )
5	1, 4	syl 17	. . . 4 |- ( A e. On -> ( -. A e. B \/ -. B e. A ) )
6	5	adantr 471	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A e. B \/ -. B e. A ) )
7		eqimss 3451	. . . . . 6 |- ( suc A = suc B -> suc A C_ suc B )
8		sucssel 5493	. . . . . 6 |- ( A e. On -> ( suc A C_ suc B -> A e. suc B ) )
9	7, 8	syl5 33	. . . . 5 |- ( A e. On -> ( suc A = suc B -> A e. suc B ) )
10		elsuci 5467	. . . . . . 7 |- ( A e. suc B -> ( A e. B \/ A = B ) )
11	10	ord 383	. . . . . 6 |- ( A e. suc B -> ( -. A e. B -> A = B ) )
12	11	com12 32	. . . . 5 |- ( -. A e. B -> ( A e. suc B -> A = B ) )
13	9, 12	syl9 73	. . . 4 |- ( A e. On -> ( -. A e. B -> ( suc A = suc B -> A = B ) ) )
14		eqimss2 3452	. . . . . 6 |- ( suc A = suc B -> suc B C_ suc A )
15		sucssel 5493	. . . . . 6 |- ( B e. On -> ( suc B C_ suc A -> B e. suc A ) )
16	14, 15	syl5 33	. . . . 5 |- ( B e. On -> ( suc A = suc B -> B e. suc A ) )
17		elsuci 5467	. . . . . . . 8 |- ( B e. suc A -> ( B e. A \/ B = A ) )
18	17	ord 383	. . . . . . 7 |- ( B e. suc A -> ( -. B e. A -> B = A ) )
19	18	com12 32	. . . . . 6 |- ( -. B e. A -> ( B e. suc A -> B = A ) )
20		eqcom 2458	. . . . . 6 |- ( B = A <-> A = B )
21	19, 20	syl6ib 234	. . . . 5 |- ( -. B e. A -> ( B e. suc A -> A = B ) )
22	16, 21	syl9 73	. . . 4 |- ( B e. On -> ( -. B e. A -> ( suc A = suc B -> A = B ) ) )
23	13, 22	jaao 516	. . 3 |- ( ( A e. On /\ B e. On ) -> ( ( -. A e. B \/ -. B e. A ) -> ( suc A = suc B -> A = B ) ) )
24	6, 23	mpd 15	. 2 |- ( ( A e. On /\ B e. On ) -> ( suc A = suc B -> A = B ) )
25		suceq 5466	. 2 |- ( A = B -> suc A = suc B )
26	24, 25	impbid1 208	1 |- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1447   e. wcel 1890   C_ wss 3371   Ord word 5400   Oncon0 5401   suc csuc 5403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pr 4611

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3014  df-sbc 3235  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-nul 3699  df-if 3849  df-sn 3936  df-pr 3938  df-op 3942  df-uni 4168  df-br 4374  df-opab 4433  df-tr 4469  df-eprel 4722  df-po 4732  df-so 4733  df-fr 4770  df-we 4772  df-ord 5404  df-on 5405  df-suc 5407

This theorem is referenced by:  peano4  6702  limenpsi  7733  fin1a2lem2  8817  bnj168  29543  sltval2  30548  sltsolem1  30562  onsuct0  31106