Theorem limsuc2 37611

Description: Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Assertion

Ref	Expression
limsuc2	|- ( ( Ord A /\ A = U. A ) -> ( B e. A <-> suc B e. A ) )

Proof of Theorem limsuc2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordunisuc2 7044	. . . . 5 |- ( Ord A -> ( A = U. A <-> A. x e. A suc x e. A ) )
2	1	biimpa 501	. . . 4 |- ( ( Ord A /\ A = U. A ) -> A. x e. A suc x e. A )
3		suceq 5790	. . . . . 6 |- ( x = B -> suc x = suc B )
4	3	eleq1d 2686	. . . . 5 |- ( x = B -> ( suc x e. A <-> suc B e. A ) )
5	4	rspccva 3308	. . . 4 |- ( ( A. x e. A suc x e. A /\ B e. A ) -> suc B e. A )
6	2, 5	sylan 488	. . 3 |- ( ( ( Ord A /\ A = U. A ) /\ B e. A ) -> suc B e. A )
7	6	ex 450	. 2 |- ( ( Ord A /\ A = U. A ) -> ( B e. A -> suc B e. A ) )
8		ordtr 5737	. . . 4 |- ( Ord A -> Tr A )
9		trsuc 5810	. . . . 5 |- ( ( Tr A /\ suc B e. A ) -> B e. A )
10	9	ex 450	. . . 4 |- ( Tr A -> ( suc B e. A -> B e. A ) )
11	8, 10	syl 17	. . 3 |- ( Ord A -> ( suc B e. A -> B e. A ) )
12	11	adantr 481	. 2 |- ( ( Ord A /\ A = U. A ) -> ( suc B e. A -> B e. A ) )
13	7, 12	impbid 202	1 |- ( ( Ord A /\ A = U. A ) -> ( B e. A <-> suc B e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   U.cuni 4436   Tr wtr 4752   Ord word 5722   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-8 1992  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  ax-un 6949

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-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  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:  aomclem4  37627