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Theorem prprc1 3736
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 3695 . 2  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 uneq1 3322 . . 3  |-  ( { A }  =  (/)  ->  ( { A }  u.  { B } )  =  ( (/)  u.  { B } ) )
3 df-pr 3647 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 uncom 3319 . . . 4  |-  ( (/)  u. 
{ B } )  =  ( { B }  u.  (/) )
5 un0 3479 . . . 4  |-  ( { B }  u.  (/) )  =  { B }
64, 5eqtr2i 2304 . . 3  |-  { B }  =  ( (/)  u.  { B } )
72, 3, 63eqtr4g 2340 . 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 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   (/)c0 3455   {csn 3640   {cpr 3641
This theorem is referenced by:  prprc2  3737  prprc  3738  prex  4217  elprchashprn2  28088  usgraedgprv  28122
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-pr 3647
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