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Theorem limensuc 7486
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
StepHypRef Expression
1 eleq1 2501 . . . 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 4782 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  suc  A  =  suc  if ( Lim 
A ,  A ,  On ) )
42, 3breq12d 4303 . . . 4  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( A 
~~  suc  A  <->  if ( Lim  A ,  A ,  On )  ~~  suc  if ( Lim  A ,  A ,  On ) ) )
51, 4imbi12d 320 . . 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 4729 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
A  <->  Lim  if ( Lim 
A ,  A ,  On ) ) )
7 limeq 4729 . . . . 5  |-  ( On  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
On 
<->  Lim  if ( Lim 
A ,  A ,  On ) ) )
8 limon 6445 . . . . 5  |-  Lim  On
96, 7, 8elimhyp 3846 . . . 4  |-  Lim  if ( Lim  A ,  A ,  On )
109limensuci 7485 . . 3  |-  ( if ( Lim  A ,  A ,  On )  e.  V  ->  if ( Lim  A ,  A ,  On )  ~~  suc  if ( Lim  A ,  A ,  On )
)
115, 10dedth 3839 . 2  |-  ( Lim 
A  ->  ( A  e.  V  ->  A  ~~  suc  A ) )
1211impcom 430 1  |-  ( ( A  e.  V  /\  Lim  A )  ->  A  ~~  suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ifcif 3789   class class class wbr 4290   Oncon0 4717   Lim wlim 4718   suc csuc 4719    ~~ cen 7305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-1o 6918  df-er 7099  df-en 7309  df-dom 7310
This theorem is referenced by:  infensuc  7487
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