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Theorem prprc2 4084
Description: A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
prprc2  |-  ( -.  B  e.  _V  ->  { A ,  B }  =  { A } )

Proof of Theorem prprc2
StepHypRef Expression
1 prcom 4051 . 2  |-  { A ,  B }  =  { B ,  A }
2 prprc1 4083 . 2  |-  ( -.  B  e.  _V  ->  { B ,  A }  =  { A } )
31, 2syl5eq 2504 1  |-  ( -.  B  e.  _V  ->  { A ,  B }  =  { A } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3068   {csn 3975   {cpr 3977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3070  df-dif 3429  df-un 3431  df-nul 3736  df-sn 3976  df-pr 3978
This theorem is referenced by:  tpprceq3  4111  prex  4632  indislem  18720  usgraedgprv  23430  indispcon  27257  1to2vfriswmgra  30736
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