ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  preqr2g Unicode version

Theorem preqr2g 3662
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the second elements are equal. Closed form of preqr2 3664. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
preqr2g  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { C ,  A }  =  { C ,  B }  ->  A  =  B ) )

Proof of Theorem preqr2g
StepHypRef Expression
1 prcom 3567 . . 3  |-  { C ,  A }  =  { A ,  C }
2 prcom 3567 . . 3  |-  { C ,  B }  =  { B ,  C }
31, 2eqeq12i 2129 . 2  |-  ( { C ,  A }  =  { C ,  B } 
<->  { A ,  C }  =  { B ,  C } )
4 preqr1g 3661 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  A  =  B ) )
53, 4syl5bi 151 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { C ,  A }  =  { C ,  B }  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   _Vcvv 2658   {cpr 3496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502
This theorem is referenced by:  opth  4127
  Copyright terms: Public domain W3C validator