Theorem limensuc 8137

Description: A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)

Assertion

Ref	Expression
limensuc	|- ( ( A e. V /\ Lim A ) -> A ~~ suc A )

Proof of Theorem limensuc

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . 4 |- ( A = if ( Lim A , A , On ) -> ( A e. V <-> if ( Lim A , A , On ) e. V ) )
2		id 22	. . . . 5 |- ( A = if ( Lim A , A , On ) -> A = if ( Lim A , A , On ) )
3		suceq 5790	. . . . 5 |- ( A = if ( Lim A , A , On ) -> suc A = suc if ( Lim A , A , On ) )
4	2, 3	breq12d 4666	. . . 4 |- ( A = if ( Lim A , A , On ) -> ( A ~~ suc A <-> if ( Lim A , A , On ) ~~ suc if ( Lim A , A , On ) ) )
5	1, 4	imbi12d 334	. . 3 |- ( A = if ( Lim A , A , On ) -> ( ( A e. V -> A ~~ suc A ) <-> ( if ( Lim A , A , On ) e. V -> if ( Lim A , A , On ) ~~ suc if ( Lim A , A , On ) ) ) )
6		limeq 5735	. . . . 5 |- ( A = if ( Lim A , A , On ) -> ( Lim A <-> Lim if ( Lim A , A , On ) ) )
7		limeq 5735	. . . . 5 |- ( On = if ( Lim A , A , On ) -> ( Lim On <-> Lim if ( Lim A , A , On ) ) )
8		limon 7036	. . . . 5 |- Lim On
9	6, 7, 8	elimhyp 4146	. . . 4 |- Lim if ( Lim A , A , On )
10	9	limensuci 8136	. . 3 |- ( if ( Lim A , A , On ) e. V -> if ( Lim A , A , On ) ~~ suc if ( Lim A , A , On ) )
11	5, 10	dedth 4139	. 2 |- ( Lim A -> ( A e. V -> A ~~ suc A ) )
12	11	impcom 446	1 |- ( ( A e. V /\ Lim A ) -> A ~~ suc A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   Oncon0 5723   Lim wlim 5724   suc csuc 5725   ~~ cen 7952

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-pow 4843  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-csb 3534  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-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957

This theorem is referenced by:  infensuc  8138