Theorem suc11 4644

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 4551	. . . . 5 |- ( A e. On -> Ord A )
2		ordn2lp 4561	. . . . . 6 |- ( Ord A -> -. ( A e. B /\ B e. A ) )
3		ianor 475	. . . . . 6 |- ( -. ( A e. B /\ B e. A ) <-> ( -. A e. B \/ -. B e. A ) )
4	2, 3	sylib 189	. . . . 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 452	. . 3 |- ( ( A e. On /\ B e. On ) -> ( -. A e. B \/ -. B e. A ) )
7		eqimss 3360	. . . . . 6 |- ( suc A = suc B -> suc A C_ suc B )
8		sucssel 4633	. . . . . 6 |- ( A e. On -> ( suc A C_ suc B -> A e. suc B ) )
9	7, 8	syl5 30	. . . . 5 |- ( A e. On -> ( suc A = suc B -> A e. suc B ) )
10		elsuci 4607	. . . . . . 7 |- ( A e. suc B -> ( A e. B \/ A = B ) )
11	10	ord 367	. . . . . 6 |- ( A e. suc B -> ( -. A e. B -> A = B ) )
12	11	com12 29	. . . . 5 |- ( -. A e. B -> ( A e. suc B -> A = B ) )
13	9, 12	syl9 68	. . . 4 |- ( A e. On -> ( -. A e. B -> ( suc A = suc B -> A = B ) ) )
14		eqimss2 3361	. . . . . 6 |- ( suc A = suc B -> suc B C_ suc A )
15		sucssel 4633	. . . . . 6 |- ( B e. On -> ( suc B C_ suc A -> B e. suc A ) )
16	14, 15	syl5 30	. . . . 5 |- ( B e. On -> ( suc A = suc B -> B e. suc A ) )
17		elsuci 4607	. . . . . . . 8 |- ( B e. suc A -> ( B e. A \/ B = A ) )
18	17	ord 367	. . . . . . 7 |- ( B e. suc A -> ( -. B e. A -> B = A ) )
19	18	com12 29	. . . . . 6 |- ( -. B e. A -> ( B e. suc A -> B = A ) )
20		eqcom 2406	. . . . . 6 |- ( B = A <-> A = B )
21	19, 20	syl6ib 218	. . . . 5 |- ( -. B e. A -> ( B e. suc A -> A = B ) )
22	16, 21	syl9 68	. . . 4 |- ( B e. On -> ( -. B e. A -> ( suc A = suc B -> A = B ) ) )
23	13, 22	jaao 496	. . 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 4606	. 2 |- ( A = B -> suc A = suc B )
26	24, 25	impbid1 195	1 |- ( ( A e. On /\ B e. On ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   = wceq 1649   e. wcel 1721   C_ wss 3280   Ord word 4540   Oncon0 4541   suc csuc 4543

This theorem is referenced by:  peano4  4826  limenpsi  7241  fin1a2lem2  8237  sltval2  25524  sltsolem1  25536  onsuct0  26095  bnj168  28803

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-suc 4547