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Theorem sucprcreg 4545
Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (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 4409 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
2 elirr 4537 . . . 4  |-  -.  A  e.  A
3 nfv 1528 . . . . 5  |-  F/ x  A  e.  A
4 eleq1 2240 . . . . 5  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
53, 4ceqsalg 2765 . . . 4  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  x  e.  A )  <->  A  e.  A ) )
62, 5mtbiri 675 . . 3  |-  ( A  e.  _V  ->  -.  A. x ( x  =  A  ->  x  e.  A ) )
7 velsn 3608 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
8 olc 711 . . . . . 6  |-  ( x  e.  { A }  ->  ( x  e.  A  \/  x  e.  { A } ) )
9 elun 3276 . . . . . . 7  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
10 ssid 3175 . . . . . . . . 9  |-  A  C_  A
11 df-suc 4368 . . . . . . . . . . 11  |-  suc  A  =  ( A  u.  { A } )
1211eqeq1i 2185 . . . . . . . . . 10  |-  ( suc 
A  =  A  <->  ( A  u.  { A } )  =  A )
13 sseq1 3178 . . . . . . . . . 10  |-  ( ( A  u.  { A } )  =  A  ->  ( ( A  u.  { A }
)  C_  A  <->  A  C_  A
) )
1412, 13sylbi 121 . . . . . . . . 9  |-  ( suc 
A  =  A  -> 
( ( A  u.  { A } )  C_  A 
<->  A  C_  A )
)
1510, 14mpbiri 168 . . . . . . . 8  |-  ( suc 
A  =  A  -> 
( A  u.  { A } )  C_  A
)
1615sseld 3154 . . . . . . 7  |-  ( suc 
A  =  A  -> 
( x  e.  ( A  u.  { A } )  ->  x  e.  A ) )
179, 16biimtrrid 153 . . . . . 6  |-  ( suc 
A  =  A  -> 
( ( x  e.  A  \/  x  e. 
{ A } )  ->  x  e.  A
) )
188, 17syl5 32 . . . . 5  |-  ( suc 
A  =  A  -> 
( x  e.  { A }  ->  x  e.  A ) )
197, 18biimtrrid 153 . . . 4  |-  ( suc 
A  =  A  -> 
( x  =  A  ->  x  e.  A
) )
2019alrimiv 1874 . . 3  |-  ( suc 
A  =  A  ->  A. x ( x  =  A  ->  x  e.  A ) )
216, 20nsyl3 626 . 2  |-  ( suc 
A  =  A  ->  -.  A  e.  _V )
221, 21impbii 126 1  |-  ( -.  A  e.  _V  <->  suc  A  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105    \/ wo 708   A.wal 1351    = wceq 1353    e. wcel 2148   _Vcvv 2737    u. cun 3127    C_ wss 3129   {csn 3591   suc csuc 4362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-suc 4368
This theorem is referenced by: (None)
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