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Theorem sucprc 4302
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 4261 . . 3  |-  suc  A  =  ( A  u.  { A } )
2 snprc 3556 . . . 4  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
3 uneq2 3192 . . . 4  |-  ( { A }  =  (/)  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
42, 3sylbi 120 . . 3  |-  ( -.  A  e.  _V  ->  ( A  u.  { A } )  =  ( A  u.  (/) ) )
51, 4syl5eq 2160 . 2  |-  ( -.  A  e.  _V  ->  suc 
A  =  ( A  u.  (/) ) )
6 un0 3364 . 2  |-  ( A  u.  (/) )  =  A
75, 6syl6eq 2164 1  |-  ( -.  A  e.  _V  ->  suc 
A  =  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1314    e. wcel 1463   _Vcvv 2658    u. cun 3037   (/)c0 3331   {csn 3495   suc csuc 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-nul 3332  df-sn 3501  df-suc 4261
This theorem is referenced by:  sucprcreg  4432  sucon  4436
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