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Theorem prprc1 3812
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
prprc1  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )

Proof of Theorem prprc1
StepHypRef Expression
1 snprc 3771 . 2  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 uneq1 3398 . . 3  |-  ( { A }  =  (/)  ->  ( { A }  u.  { B } )  =  ( (/)  u.  { B } ) )
3 df-pr 3723 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 uncom 3395 . . . 4  |-  ( (/)  u. 
{ B } )  =  ( { B }  u.  (/) )
5 un0 3555 . . . 4  |-  ( { B }  u.  (/) )  =  { B }
64, 5eqtr2i 2379 . . 3  |-  { B }  =  ( (/)  u.  { B } )
72, 3, 63eqtr4g 2415 . 2  |-  ( { A }  =  (/)  ->  { A ,  B }  =  { B } )
81, 7sylbi 187 1  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1642    e. wcel 1710   _Vcvv 2864    u. cun 3226   (/)c0 3531   {csn 3716   {cpr 3717
This theorem is referenced by:  prprc2  3813  prprc  3814  prex  4296  elprchashprn2  27464  usgraedgprv  27543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-v 2866  df-dif 3231  df-un 3233  df-nul 3532  df-sn 3722  df-pr 3723
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