MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prprc2 Structured version   Unicode version

Theorem prprc2 4127
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 4094 . 2  |-  { A ,  B }  =  { B ,  A }
2 prprc1 4126 . 2  |-  ( -.  B  e.  _V  ->  { B ,  A }  =  { A } )
31, 2syl5eq 2507 1  |-  ( -.  B  e.  _V  ->  { A ,  B }  =  { A } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-nul 3784  df-sn 4017  df-pr 4019
This theorem is referenced by:  tpprceq3  4156  prex  4679  indislem  19671  usgraedgprv  24581  1to2vfriswmgra  25211  indispcon  28946
  Copyright terms: Public domain W3C validator