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Theorem limensuc 7081
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 2376 . . . 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 4494 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  suc  A  =  suc  if ( Lim 
A ,  A ,  On ) )
42, 3breq12d 4073 . . . 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 4441 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
A  <->  Lim  if ( Lim 
A ,  A ,  On ) ) )
7 limeq 4441 . . . . 5  |-  ( On  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
On 
<->  Lim  if ( Lim 
A ,  A ,  On ) ) )
8 limon 4664 . . . . 5  |-  Lim  On
96, 7, 8elimhyp 3647 . . . 4  |-  Lim  if ( Lim  A ,  A ,  On )
109limensuci 7080 . . 3  |-  ( if ( Lim  A ,  A ,  On )  e.  V  ->  if ( Lim  A ,  A ,  On )  ~~  suc  if ( Lim  A ,  A ,  On )
)
115, 10dedth 3640 . 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 1633    e. wcel 1701   ifcif 3599   class class class wbr 4060   Oncon0 4429   Lim wlim 4430   suc csuc 4431    ~~ cen 6903
This theorem is referenced by:  infensuc  7082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-1o 6521  df-er 6702  df-en 6907  df-dom 6908
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