Theorem suc11 4981

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 4888	. . . . 5 |- ( A e. On -> Ord A )
2		ordn2lp 4898	. . . . . 6 |- ( Ord A -> -. ( A e. B /\ B e. A ) )
3		ianor 488	. . . . . 6 |- ( -. ( A e. B /\ B e. A ) <-> ( -. A e. B \/ -. B e. A ) )
4	2, 3	sylib 196	. . . . 5 |- ( Ord A -> ( -. A e. B \/ -. B e. A ) )
5	1, 4	syl 16	. . . 4 |- ( A e. On -> ( -. A e. B \/ -. B e. A ) )
6	5	adantr 465	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A e. B \/ -. B e. A ) )
7		eqimss 3556	. . . . . 6 |- ( suc A = suc B -> suc A C_ suc B )
8		sucssel 4970	. . . . . 6 |- ( A e. On -> ( suc A C_ suc B -> A e. suc B ) )
9	7, 8	syl5 32	. . . . 5 |- ( A e. On -> ( suc A = suc B -> A e. suc B ) )
10		elsuci 4944	. . . . . . 7 |- ( A e. suc B -> ( A e. B \/ A = B ) )
11	10	ord 377	. . . . . 6 |- ( A e. suc B -> ( -. A e. B -> A = B ) )
12	11	com12 31	. . . . 5 |- ( -. A e. B -> ( A e. suc B -> A = B ) )
13	9, 12	syl9 71	. . . 4 |- ( A e. On -> ( -. A e. B -> ( suc A = suc B -> A = B ) ) )
14		eqimss2 3557	. . . . . 6 |- ( suc A = suc B -> suc B C_ suc A )
15		sucssel 4970	. . . . . 6 |- ( B e. On -> ( suc B C_ suc A -> B e. suc A ) )
16	14, 15	syl5 32	. . . . 5 |- ( B e. On -> ( suc A = suc B -> B e. suc A ) )
17		elsuci 4944	. . . . . . . 8 |- ( B e. suc A -> ( B e. A \/ B = A ) )
18	17	ord 377	. . . . . . 7 |- ( B e. suc A -> ( -. B e. A -> B = A ) )
19	18	com12 31	. . . . . 6 |- ( -. B e. A -> ( B e. suc A -> B = A ) )
20		eqcom 2476	. . . . . 6 |- ( B = A <-> A = B )
21	19, 20	syl6ib 226	. . . . 5 |- ( -. B e. A -> ( B e. suc A -> A = B ) )
22	16, 21	syl9 71	. . . 4 |- ( B e. On -> ( -. B e. A -> ( suc A = suc B -> A = B ) ) )
23	13, 22	jaao 509	. . 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 4943	. 2 |- ( A = B -> suc A = suc B )
26	24, 25	impbid1 203	1 |- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1379   e. wcel 1767   C_ wss 3476   Ord word 4877   Oncon0 4878   suc csuc 4880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884

This theorem is referenced by:  peano4  6706  limenpsi  7692  fin1a2lem2  8781  sltval2  29021  sltsolem1  29033  onsuct0  29511  bnj168  32883