MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  preqr2 Structured version   Unicode version

Theorem preqr2 3965
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same first element, the second elements are equal. (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
preqr2.1  |-  A  e. 
_V
preqr2.2  |-  B  e. 
_V
Assertion
Ref Expression
preqr2  |-  ( { C ,  A }  =  { C ,  B }  ->  A  =  B )

Proof of Theorem preqr2
StepHypRef Expression
1 prcom 3874 . . 3  |-  { C ,  A }  =  { A ,  C }
2 prcom 3874 . . 3  |-  { C ,  B }  =  { B ,  C }
31, 2eqeq12i 2448 . 2  |-  ( { C ,  A }  =  { C ,  B } 
<->  { A ,  C }  =  { B ,  C } )
4 preqr2.1 . . 3  |-  A  e. 
_V
5 preqr2.2 . . 3  |-  B  e. 
_V
64, 5preqr1 3964 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  A  =  B )
73, 6sylbi 188 1  |-  ( { C ,  A }  =  { C ,  B }  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   {cpr 3807
This theorem is referenced by:  preq12b  3966  opth  4427  opthreg  7565  usgra2edg  21394  usgraedgreu  21399  nbgraf1olem5  21447  altopthsn  25798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-sn 3812  df-pr 3813
  Copyright terms: Public domain W3C validator