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

Theorem preqr2 3916
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 3826 . . 3  |-  { C ,  A }  =  { A ,  C }
2 prcom 3826 . . 3  |-  { C ,  B }  =  { B ,  C }
31, 2eqeq12i 2401 . 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 3915 . 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 1649    e. wcel 1717   _Vcvv 2900   {cpr 3759
This theorem is referenced by:  preq12b  3917  opth  4377  opthreg  7507  usgra2edg  21269  usgraedgreu  21274  nbgraf1olem5  21322  altopthsn  25521
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-un 3269  df-sn 3764  df-pr 3765
  Copyright terms: Public domain W3C validator