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Theorem sucprc 4590
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 4521 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 snprc 3807 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
3 uneq2 3431 . . . 4  |-  ( { A }  =  (/)  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
42, 3sylbi 188 . . 3  |-  ( -.  A  e.  _V  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
51, 4syl5eq 2424 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  ( A  u.  (/) ) )
6 un0 3588 . 2  |-  ( A  u.  (/) )  =  A
75, 6syl6eq 2428 1  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2892    u. cun 3254   (/)c0 3564   {csn 3750   suc csuc 4517
This theorem is referenced by:  nsuceq0  4595  sucon  4721  ordsuc  4727  sucprcreg  7493  suc11reg  7500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-v 2894  df-dif 3259  df-un 3261  df-nul 3565  df-sn 3756  df-suc 4521
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