Theorem suc11 4827

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 4734	. . . . 5 |- ( A e. On -> Ord A )
2		ordn2lp 4744	. . . . . 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 3413	. . . . . 6 |- ( suc A = suc B -> suc A C_ suc B )
8		sucssel 4816	. . . . . 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 4790	. . . . . . 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 3414	. . . . . 6 |- ( suc A = suc B -> suc B C_ suc A )
15		sucssel 4816	. . . . . 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 4790	. . . . . . . 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 2445	. . . . . 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 4789	. 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 1369   e. wcel 1756   C_ wss 3333   Ord word 4723   Oncon0 4724   suc csuc 4726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-tr 4391  df-eprel 4637  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-suc 4730

This theorem is referenced by:  peano4  6503  limenpsi  7491  fin1a2lem2  8575  sltval2  27802  sltsolem1  27814  onsuct0  28292  bnj168  31726