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Theorem preqr1g 3566
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 3568. (Contributed by Jim Kingdon, 21-Sep-2018.)
Assertion
Ref Expression
preqr1g  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  A  =  B ) )

Proof of Theorem preqr1g
StepHypRef Expression
1 prid1g 3504 . . . . . . 7  |-  ( A  e.  _V  ->  A  e.  { A ,  C } )
2 eleq2 2143 . . . . . . 7  |-  ( { A ,  C }  =  { B ,  C }  ->  ( A  e. 
{ A ,  C } 
<->  A  e.  { B ,  C } ) )
31, 2syl5ibcom 153 . . . . . 6  |-  ( A  e.  _V  ->  ( { A ,  C }  =  { B ,  C }  ->  A  e.  { B ,  C }
) )
4 elprg 3426 . . . . . 6  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
53, 4sylibd 147 . . . . 5  |-  ( A  e.  _V  ->  ( { A ,  C }  =  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) ) )
65adantr 270 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) ) )
76imp 122 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  { A ,  C }  =  { B ,  C } )  -> 
( A  =  B  \/  A  =  C ) )
8 prid1g 3504 . . . . . . 7  |-  ( B  e.  _V  ->  B  e.  { B ,  C } )
9 eleq2 2143 . . . . . . 7  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  e. 
{ A ,  C } 
<->  B  e.  { B ,  C } ) )
108, 9syl5ibrcom 155 . . . . . 6  |-  ( B  e.  _V  ->  ( { A ,  C }  =  { B ,  C }  ->  B  e.  { A ,  C }
) )
11 elprg 3426 . . . . . 6  |-  ( B  e.  _V  ->  ( B  e.  { A ,  C }  <->  ( B  =  A  \/  B  =  C ) ) )
1210, 11sylibd 147 . . . . 5  |-  ( B  e.  _V  ->  ( { A ,  C }  =  { B ,  C }  ->  ( B  =  A  \/  B  =  C ) ) )
1312adantl 271 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  ( B  =  A  \/  B  =  C ) ) )
1413imp 122 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  { A ,  C }  =  { B ,  C } )  -> 
( B  =  A  \/  B  =  C ) )
15 eqcom 2084 . . 3  |-  ( A  =  B  <->  B  =  A )
16 eqeq2 2091 . . 3  |-  ( A  =  C  ->  ( B  =  A  <->  B  =  C ) )
177, 14, 15, 16oplem1 917 . 2  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  { A ,  C }  =  { B ,  C } )  ->  A  =  B )
1817ex 113 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 662    = wceq 1285    e. wcel 1434   _Vcvv 2602   {cpr 3407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413
This theorem is referenced by:  preqr2g  3567
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