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Theorem limensuc 7687
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 2526 . . . 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 4932 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  suc  A  =  suc  if ( Lim 
A ,  A ,  On ) )
42, 3breq12d 4452 . . . 4  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( A 
~~  suc  A  <->  if ( Lim  A ,  A ,  On )  ~~  suc  if ( Lim  A ,  A ,  On ) ) )
51, 4imbi12d 318 . . 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 4879 . . . . 5  |-  ( A  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
A  <->  Lim  if ( Lim 
A ,  A ,  On ) ) )
7 limeq 4879 . . . . 5  |-  ( On  =  if ( Lim 
A ,  A ,  On )  ->  ( Lim 
On 
<->  Lim  if ( Lim 
A ,  A ,  On ) ) )
8 limon 6644 . . . . 5  |-  Lim  On
96, 7, 8elimhyp 3987 . . . 4  |-  Lim  if ( Lim  A ,  A ,  On )
109limensuci 7686 . . 3  |-  ( if ( Lim  A ,  A ,  On )  e.  V  ->  if ( Lim  A ,  A ,  On )  ~~  suc  if ( Lim  A ,  A ,  On )
)
115, 10dedth 3980 . 2  |-  ( Lim 
A  ->  ( A  e.  V  ->  A  ~~  suc  A ) )
1211impcom 428 1  |-  ( ( A  e.  V  /\  Lim  A )  ->  A  ~~  suc  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   ifcif 3929   class class class wbr 4439   Oncon0 4867   Lim wlim 4868   suc csuc 4869    ~~ cen 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511
This theorem is referenced by:  infensuc  7688
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