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Theorem limensuc 7038
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 2343 . . . 4  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( A  e.  V  <->  if ( Lim  A ,  A ,  On )  e.  V
) )
2 id 19 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  A  =  if ( Lim  A ,  A ,  On ) )
3 suceq 4457 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  suc  A  =  suc  if ( Lim 
A ,  A ,  On ) )
42, 3breq12d 4036 . . . 4  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( A 
~~  suc  A  <->  if ( Lim  A ,  A ,  On )  ~~  suc  if ( Lim  A ,  A ,  On ) ) )
51, 4imbi12d 311 . . 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 4404 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
A  <->  Lim  if ( Lim 
A ,  A ,  On ) ) )
7 limeq 4404 . . . . 5  |-  ( On  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
On 
<->  Lim  if ( Lim 
A ,  A ,  On ) ) )
8 limon 4627 . . . . 5  |-  Lim  On
96, 7, 8elimhyp 3613 . . . 4  |-  Lim  if ( Lim  A ,  A ,  On )
109limensuci 7037 . . 3  |-  ( if ( Lim  A ,  A ,  On )  e.  V  ->  if ( Lim  A ,  A ,  On )  ~~  suc  if ( Lim  A ,  A ,  On )
)
115, 10dedth 3606 . 2  |-  ( Lim 
A  ->  ( A  e.  V  ->  A  ~~  suc  A ) )
1211impcom 419 1  |-  ( ( A  e.  V  /\  Lim  A )  ->  A  ~~  suc  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ifcif 3565   class class class wbr 4023   Oncon0 4392   Lim wlim 4393   suc csuc 4394    ~~ cen 6860
This theorem is referenced by:  infensuc  7039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865
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