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Theorem sucprcreg 7329
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 4483 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
2 elirr 7328 . . . 4  |-  -.  A  e.  A
3 nfv 1609 . . . . 5  |-  F/ x  A  e.  A
4 eleq1 2356 . . . . 5  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
53, 4ceqsalg 2825 . . . 4  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  x  e.  A )  <->  A  e.  A ) )
62, 5mtbiri 294 . . 3  |-  ( A  e.  _V  ->  -.  A. x ( x  =  A  ->  x  e.  A ) )
7 elsn 3668 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
8 olc 373 . . . . . 6  |-  ( x  e.  { A }  ->  ( x  e.  A  \/  x  e.  { A } ) )
9 elun 3329 . . . . . . 7  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
10 ssid 3210 . . . . . . . . 9  |-  A  C_  A
11 df-suc 4414 . . . . . . . . . . 11  |-  suc  A  =  ( A  u.  { A } )
1211eqeq1i 2303 . . . . . . . . . 10  |-  ( suc 
A  =  A  <->  ( A  u.  { A } )  =  A )
13 sseq1 3212 . . . . . . . . . 10  |-  ( ( A  u.  { A } )  =  A  ->  ( ( A  u.  { A }
)  C_  A  <->  A  C_  A
) )
1412, 13sylbi 187 . . . . . . . . 9  |-  ( suc 
A  =  A  -> 
( ( A  u.  { A } )  C_  A 
<->  A  C_  A )
)
1510, 14mpbiri 224 . . . . . . . 8  |-  ( suc 
A  =  A  -> 
( A  u.  { A } )  C_  A
)
1615sseld 3192 . . . . . . 7  |-  ( suc 
A  =  A  -> 
( x  e.  ( A  u.  { A } )  ->  x  e.  A ) )
179, 16syl5bir 209 . . . . . 6  |-  ( suc 
A  =  A  -> 
( ( x  e.  A  \/  x  e. 
{ A } )  ->  x  e.  A
) )
188, 17syl5 28 . . . . 5  |-  ( suc 
A  =  A  -> 
( x  e.  { A }  ->  x  e.  A ) )
197, 18syl5bir 209 . . . 4  |-  ( suc 
A  =  A  -> 
( x  =  A  ->  x  e.  A
) )
2019alrimiv 1621 . . 3  |-  ( suc 
A  =  A  ->  A. x ( x  =  A  ->  x  e.  A ) )
216, 20nsyl3 111 . 2  |-  ( suc 
A  =  A  ->  -.  A  e.  _V )
221, 21impbii 180 1  |-  ( -.  A  e.  _V  <->  suc  A  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357   A.wal 1530    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    C_ wss 3165   {csn 3653   suc csuc 4410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-reg 7322
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-sn 3659  df-pr 3660  df-suc 4414
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