Theorem suc11 4970

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 4877	. . . . 5 |- ( A e. On -> Ord A )
2		ordn2lp 4887	. . . . . 6 |- ( Ord A -> -. ( A e. B /\ B e. A ) )
3		ianor 486	. . . . . 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 463	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A e. B \/ -. B e. A ) )
7		eqimss 3541	. . . . . 6 |- ( suc A = suc B -> suc A C_ suc B )
8		sucssel 4959	. . . . . 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 4933	. . . . . . 7 |- ( A e. suc B -> ( A e. B \/ A = B ) )
11	10	ord 375	. . . . . 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 3542	. . . . . 6 |- ( suc A = suc B -> suc B C_ suc A )
15		sucssel 4959	. . . . . 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 4933	. . . . . . . 8 |- ( B e. suc A -> ( B e. A \/ B = A ) )
18	17	ord 375	. . . . . . 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 2463	. . . . . 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 507	. . 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 4932	. 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 366   /\ wa 367   = wceq 1398   e. wcel 1823   C_ wss 3461   Ord word 4866   Oncon0 4867   suc csuc 4869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873

This theorem is referenced by:  peano4  6695  limenpsi  7685  fin1a2lem2  8772  sltval2  29656  sltsolem1  29668  onsuct0  30134  bnj168  34186