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Theorem sucprcreg 7569
Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity). (Contributed by NM, 9-Jul-2004.)
Assertion
Ref Expression
sucprcreg  |-  ( -.  A  e.  _V  <->  suc  A  =  A )

Proof of Theorem sucprcreg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sucprc 4658 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
2 elirr 7568 . . . 4  |-  -.  A  e.  A
3 nfv 1630 . . . . 5  |-  F/ x  A  e.  A
4 eleq1 2498 . . . . 5  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
53, 4ceqsalg 2982 . . . 4  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  x  e.  A )  <->  A  e.  A ) )
62, 5mtbiri 296 . . 3  |-  ( A  e.  _V  ->  -.  A. x ( x  =  A  ->  x  e.  A ) )
7 elsn 3831 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
8 olc 375 . . . . . 6  |-  ( x  e.  { A }  ->  ( x  e.  A  \/  x  e.  { A } ) )
9 elun 3490 . . . . . . 7  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
10 ssid 3369 . . . . . . . . 9  |-  A  C_  A
11 df-suc 4589 . . . . . . . . . . 11  |-  suc  A  =  ( A  u.  { A } )
1211eqeq1i 2445 . . . . . . . . . 10  |-  ( suc 
A  =  A  <->  ( A  u.  { A } )  =  A )
13 sseq1 3371 . . . . . . . . . 10  |-  ( ( A  u.  { A } )  =  A  ->  ( ( A  u.  { A }
)  C_  A  <->  A  C_  A
) )
1412, 13sylbi 189 . . . . . . . . 9  |-  ( suc 
A  =  A  -> 
( ( A  u.  { A } )  C_  A 
<->  A  C_  A )
)
1510, 14mpbiri 226 . . . . . . . 8  |-  ( suc 
A  =  A  -> 
( A  u.  { A } )  C_  A
)
1615sseld 3349 . . . . . . 7  |-  ( suc 
A  =  A  -> 
( x  e.  ( A  u.  { A } )  ->  x  e.  A ) )
179, 16syl5bir 211 . . . . . 6  |-  ( suc 
A  =  A  -> 
( ( x  e.  A  \/  x  e. 
{ A } )  ->  x  e.  A
) )
188, 17syl5 31 . . . . 5  |-  ( suc 
A  =  A  -> 
( x  e.  { A }  ->  x  e.  A ) )
197, 18syl5bir 211 . . . 4  |-  ( suc 
A  =  A  -> 
( x  =  A  ->  x  e.  A
) )
2019alrimiv 1642 . . 3  |-  ( suc 
A  =  A  ->  A. x ( x  =  A  ->  x  e.  A ) )
216, 20nsyl3 114 . 2  |-  ( suc 
A  =  A  ->  -.  A  e.  _V )
221, 21impbii 182 1  |-  ( -.  A  e.  _V  <->  suc  A  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359   A.wal 1550    = wceq 1653    e. wcel 1726   _Vcvv 2958    u. cun 3320    C_ wss 3322   {csn 3816   suc csuc 4585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-reg 7562
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-sn 3822  df-pr 3823  df-suc 4589
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