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Theorem prprc1 4082
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 15-Jul-1993.)
Assertion
Ref Expression
prprc1  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )

Proof of Theorem prprc1
StepHypRef Expression
1 snprc 4035 . 2  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
2 uneq1 3590 . . 3  |-  ( { A }  =  (/)  ->  ( { A }  u.  { B } )  =  ( (/)  u.  { B } ) )
3 df-pr 3975 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
4 uncom 3587 . . . 4  |-  ( (/)  u. 
{ B } )  =  ( { B }  u.  (/) )
5 un0 3764 . . . 4  |-  ( { B }  u.  (/) )  =  { B }
64, 5eqtr2i 2432 . . 3  |-  { B }  =  ( (/)  u.  { B } )
72, 3, 63eqtr4g 2468 . 2  |-  ( { A }  =  (/)  ->  { A ,  B }  =  { B } )
81, 7sylbi 195 1  |-  ( -.  A  e.  _V  ->  { A ,  B }  =  { B } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1405    e. wcel 1842   _Vcvv 3059    u. cun 3412   (/)c0 3738   {csn 3972   {cpr 3974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3061  df-dif 3417  df-un 3419  df-nul 3739  df-sn 3973  df-pr 3975
This theorem is referenced by:  prprc2  4083  prprc  4084  prex  4633  elprchashprn2  12510  usgraedgprv  24793
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