Theorem limsuc 7049

Description: The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.)

Assertion

Ref	Expression
limsuc	|- ( Lim A -> ( B e. A <-> suc B e. A ) )

Proof of Theorem limsuc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dflim4 7048	. . 3 |- ( Lim A <-> ( Ord A /\ (/) e. A /\ A. x e. A suc x e. A ) )
2		suceq 5790	. . . . . 6 |- ( x = B -> suc x = suc B )
3	2	eleq1d 2686	. . . . 5 |- ( x = B -> ( suc x e. A <-> suc B e. A ) )
4	3	rspccv 3306	. . . 4 |- ( A. x e. A suc x e. A -> ( B e. A -> suc B e. A ) )
5	4	3ad2ant3 1084	. . 3 |- ( ( Ord A /\ (/) e. A /\ A. x e. A suc x e. A ) -> ( B e. A -> suc B e. A ) )
6	1, 5	sylbi 207	. 2 |- ( Lim A -> ( B e. A -> suc B e. A ) )
7		limord 5784	. . 3 |- ( Lim A -> Ord A )
8		ordtr 5737	. . 3 |- ( Ord A -> Tr A )
9		trsuc 5810	. . . 4 |- ( ( Tr A /\ suc B e. A ) -> B e. A )
10	9	ex 450	. . 3 |- ( Tr A -> ( suc B e. A -> B e. A ) )
11	7, 8, 10	3syl 18	. 2 |- ( Lim A -> ( suc B e. A -> B e. A ) )
12	6, 11	impbid 202	1 |- ( Lim A -> ( B e. A <-> suc B e. A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   Tr wtr 4752   Ord word 5722   Lim wlim 5724   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-lim 5728  df-suc 5729

This theorem is referenced by:  limsssuc  7050  limuni3  7052  peano2b  7081  rdgsucg  7519  rdgsucmptnf  7525  oesuclem  7605  oaordi  7626  omordi  7646  oeordi  7667  oelim2  7675  limenpsi  8135  r1tr  8639  r1ordg  8641  r1pwss  8647  r1val1  8649  rankdmr1  8664  rankr1bg  8666  pwwf  8670  rankr1c  8684  rankonidlem  8691  ranklim  8707  r1pwcl  8710  rankxplim3  8744  infxpenlem  8836  alephordi  8897  cflm  9072  cfslb2n  9090  alephreg  9404  r1limwun  9558  rankcf  9599  inatsk  9600