Theorem suc11 5831

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 5733	. . . . 5 |- ( A e. On -> Ord A )
2		ordn2lp 5743	. . . . 5 |- ( Ord A -> -. ( A e. B /\ B e. A ) )
3		pm3.13 522	. . . . 5 |- ( -. ( A e. B /\ B e. A ) -> ( -. A e. B \/ -. B e. A ) )
4	1, 2, 3	3syl 18	. . . 4 |- ( A e. On -> ( -. A e. B \/ -. B e. A ) )
5	4	adantr 481	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A e. B \/ -. B e. A ) )
6		eqimss 3657	. . . . . 6 |- ( suc A = suc B -> suc A C_ suc B )
7		sucssel 5819	. . . . . 6 |- ( A e. On -> ( suc A C_ suc B -> A e. suc B ) )
8	6, 7	syl5 34	. . . . 5 |- ( A e. On -> ( suc A = suc B -> A e. suc B ) )
9		elsuci 5791	. . . . . . 7 |- ( A e. suc B -> ( A e. B \/ A = B ) )
10	9	ord 392	. . . . . 6 |- ( A e. suc B -> ( -. A e. B -> A = B ) )
11	10	com12 32	. . . . 5 |- ( -. A e. B -> ( A e. suc B -> A = B ) )
12	8, 11	syl9 77	. . . 4 |- ( A e. On -> ( -. A e. B -> ( suc A = suc B -> A = B ) ) )
13		eqimss2 3658	. . . . . 6 |- ( suc A = suc B -> suc B C_ suc A )
14		sucssel 5819	. . . . . 6 |- ( B e. On -> ( suc B C_ suc A -> B e. suc A ) )
15	13, 14	syl5 34	. . . . 5 |- ( B e. On -> ( suc A = suc B -> B e. suc A ) )
16		elsuci 5791	. . . . . . . 8 |- ( B e. suc A -> ( B e. A \/ B = A ) )
17	16	ord 392	. . . . . . 7 |- ( B e. suc A -> ( -. B e. A -> B = A ) )
18		eqcom 2629	. . . . . . 7 |- ( B = A <-> A = B )
19	17, 18	syl6ib 241	. . . . . 6 |- ( B e. suc A -> ( -. B e. A -> A = B ) )
20	19	com12 32	. . . . 5 |- ( -. B e. A -> ( B e. suc A -> A = B ) )
21	15, 20	syl9 77	. . . 4 |- ( B e. On -> ( -. B e. A -> ( suc A = suc B -> A = B ) ) )
22	12, 21	jaao 531	. . 3 |- ( ( A e. On /\ B e. On ) -> ( ( -. A e. B \/ -. B e. A ) -> ( suc A = suc B -> A = B ) ) )
23	5, 22	mpd 15	. 2 |- ( ( A e. On /\ B e. On ) -> ( suc A = suc B -> A = B ) )
24		suceq 5790	. 2 |- ( A = B -> suc A = suc B )
25	23, 24	impbid1 215	1 |- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   Ord word 5722   Oncon0 5723   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-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

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-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  peano4  7088  limenpsi  8135  fin1a2lem2  9223  bnj168  30798  sltval2  31809  sltsolem1  31826  nosepnelem  31830  nolt02o  31845  onsuct0  32440