Theorem ordsssuc2 5814

Description: An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
ordsssuc2	|- ( ( Ord A /\ B e. On ) -> ( A C_ B <-> A e. suc B ) )

Proof of Theorem ordsssuc2

Step	Hyp	Ref	Expression
1		elong 5731	. . . . 5 |- ( A e. _V -> ( A e. On <-> Ord A ) )
2	1	biimprd 238	. . . 4 |- ( A e. _V -> ( Ord A -> A e. On ) )
3	2	anim1d 588	. . 3 |- ( A e. _V -> ( ( Ord A /\ B e. On ) -> ( A e. On /\ B e. On ) ) )
4		onsssuc 5813	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A C_ B <-> A e. suc B ) )
5	3, 4	syl6 35	. 2 |- ( A e. _V -> ( ( Ord A /\ B e. On ) -> ( A C_ B <-> A e. suc B ) ) )
6		annim 441	. . . . 5 |- ( ( B e. On /\ -. A e. _V ) <-> -. ( B e. On -> A e. _V ) )
7		ssexg 4804	. . . . . . 7 |- ( ( A C_ B /\ B e. On ) -> A e. _V )
8	7	ex 450	. . . . . 6 |- ( A C_ B -> ( B e. On -> A e. _V ) )
9		elex 3212	. . . . . . 7 |- ( A e. suc B -> A e. _V )
10	9	a1d 25	. . . . . 6 |- ( A e. suc B -> ( B e. On -> A e. _V ) )
11	8, 10	pm5.21ni 367	. . . . 5 |- ( -. ( B e. On -> A e. _V ) -> ( A C_ B <-> A e. suc B ) )
12	6, 11	sylbi 207	. . . 4 |- ( ( B e. On /\ -. A e. _V ) -> ( A C_ B <-> A e. suc B ) )
13	12	expcom 451	. . 3 |- ( -. A e. _V -> ( B e. On -> ( A C_ B <-> A e. suc B ) ) )
14	13	adantld 483	. 2 |- ( -. A e. _V -> ( ( Ord A /\ B e. On ) -> ( A C_ B <-> A e. suc B ) ) )
15	5, 14	pm2.61i 176	1 |- ( ( Ord A /\ B e. On ) -> ( A C_ B <-> A e. suc B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   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-3or 1038  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-pss 3590  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:  ordunisuc2  7044