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Theorem sucprc 4467
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 4398 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 snprc 3695 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
3 uneq2 3323 . . . 4  |-  ( { A }  =  (/)  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
42, 3sylbi 187 . . 3  |-  ( -.  A  e.  _V  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
51, 4syl5eq 2327 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  ( A  u.  (/) ) )
6 un0 3479 . 2  |-  ( A  u.  (/) )  =  A
75, 6syl6eq 2331 1  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   (/)c0 3455   {csn 3640   suc csuc 4394
This theorem is referenced by:  nsuceq0  4472  sucon  4599  ordsuc  4605  sucprcreg  7313  suc11reg  7320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-sn 3646  df-suc 4398
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