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Theorem sucprc 4195
Description: A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
Assertion
Ref Expression
sucprc  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )

Proof of Theorem sucprc
StepHypRef Expression
1 df-suc 4154 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 snprc 3475 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
3 uneq2 3130 . . . 4  |-  ( { A }  =  (/)  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
42, 3sylbi 119 . . 3  |-  ( -.  A  e.  _V  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
51, 4syl5eq 2127 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  ( A  u.  (/) ) )
6 un0 3294 . 2  |-  ( A  u.  (/) )  =  A
75, 6syl6eq 2131 1  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1285    e. wcel 1434   _Vcvv 2610    u. cun 2980   (/)c0 3267   {csn 3416   suc csuc 4148
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984  df-un 2986  df-nul 3268  df-sn 3422  df-suc 4154
This theorem is referenced by:  sucprcreg  4320  sucon  4324
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