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Theorem sucprc 4483
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 4414 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 snprc 3708 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
3 uneq2 3336 . . . 4  |-  ( { A }  =  (/)  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
42, 3sylbi 187 . . 3  |-  ( -.  A  e.  _V  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
51, 4syl5eq 2340 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  ( A  u.  (/) ) )
6 un0 3492 . 2  |-  ( A  u.  (/) )  =  A
75, 6syl6eq 2344 1  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163   (/)c0 3468   {csn 3653   suc csuc 4410
This theorem is referenced by:  nsuceq0  4488  sucon  4615  ordsuc  4621  sucprcreg  7329  suc11reg  7336
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-suc 4414
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