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Theorem limensuc 7684
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 2532 . . . 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 4936 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  suc  A  =  suc  if ( Lim 
A ,  A ,  On ) )
42, 3breq12d 4453 . . . 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 4883 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
A  <->  Lim  if ( Lim 
A ,  A ,  On ) ) )
7 limeq 4883 . . . . 5  |-  ( On  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
On 
<->  Lim  if ( Lim 
A ,  A ,  On ) ) )
8 limon 6642 . . . . 5  |-  Lim  On
96, 7, 8elimhyp 3991 . . . 4  |-  Lim  if ( Lim  A ,  A ,  On )
109limensuci 7683 . . 3  |-  ( if ( Lim  A ,  A ,  On )  e.  V  ->  if ( Lim  A ,  A ,  On )  ~~  suc  if ( Lim  A ,  A ,  On )
)
115, 10dedth 3984 . 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 1374    e. wcel 1762   ifcif 3932   class class class wbr 4440   Oncon0 4871   Lim wlim 4872   suc csuc 4873    ~~ cen 7503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508
This theorem is referenced by:  infensuc  7685
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