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Theorem sucprcreg 4393
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 4263 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
2 elirr 4385 . . . 4  |-  -.  A  e.  A
3 nfv 1473 . . . . 5  |-  F/ x  A  e.  A
4 eleq1 2157 . . . . 5  |-  ( x  =  A  ->  (
x  e.  A  <->  A  e.  A ) )
53, 4ceqsalg 2661 . . . 4  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  x  e.  A )  <->  A  e.  A ) )
62, 5mtbiri 638 . . 3  |-  ( A  e.  _V  ->  -.  A. x ( x  =  A  ->  x  e.  A ) )
7 velsn 3483 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
8 olc 670 . . . . . 6  |-  ( x  e.  { A }  ->  ( x  e.  A  \/  x  e.  { A } ) )
9 elun 3156 . . . . . . 7  |-  ( x  e.  ( A  u.  { A } )  <->  ( x  e.  A  \/  x  e.  { A } ) )
10 ssid 3059 . . . . . . . . 9  |-  A  C_  A
11 df-suc 4222 . . . . . . . . . . 11  |-  suc  A  =  ( A  u.  { A } )
1211eqeq1i 2102 . . . . . . . . . 10  |-  ( suc 
A  =  A  <->  ( A  u.  { A } )  =  A )
13 sseq1 3062 . . . . . . . . . 10  |-  ( ( A  u.  { A } )  =  A  ->  ( ( A  u.  { A }
)  C_  A  <->  A  C_  A
) )
1412, 13sylbi 120 . . . . . . . . 9  |-  ( suc 
A  =  A  -> 
( ( A  u.  { A } )  C_  A 
<->  A  C_  A )
)
1510, 14mpbiri 167 . . . . . . . 8  |-  ( suc 
A  =  A  -> 
( A  u.  { A } )  C_  A
)
1615sseld 3038 . . . . . . 7  |-  ( suc 
A  =  A  -> 
( x  e.  ( A  u.  { A } )  ->  x  e.  A ) )
179, 16syl5bir 152 . . . . . 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, 18syl5bir 152 . . . 4  |-  ( suc 
A  =  A  -> 
( x  =  A  ->  x  e.  A
) )
2019alrimiv 1809 . . 3  |-  ( suc 
A  =  A  ->  A. x ( x  =  A  ->  x  e.  A ) )
216, 20nsyl3 594 . 2  |-  ( suc 
A  =  A  ->  -.  A  e.  _V )
221, 21impbii 125 1  |-  ( -.  A  e.  _V  <->  suc  A  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ wo 667   A.wal 1294    = wceq 1296    e. wcel 1445   _Vcvv 2633    u. cun 3011    C_ wss 3013   {csn 3466   suc csuc 4216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-suc 4222
This theorem is referenced by: (None)
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