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Theorem preqr1g 3661
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 3663. (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 3595 . . . . . . 7  |-  ( A  e.  _V  ->  A  e.  { A ,  C } )
2 eleq2 2179 . . . . . . 7  |-  ( { A ,  C }  =  { B ,  C }  ->  ( A  e. 
{ A ,  C } 
<->  A  e.  { B ,  C } ) )
31, 2syl5ibcom 154 . . . . . 6  |-  ( A  e.  _V  ->  ( { A ,  C }  =  { B ,  C }  ->  A  e.  { B ,  C }
) )
4 elprg 3515 . . . . . 6  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
53, 4sylibd 148 . . . . 5  |-  ( A  e.  _V  ->  ( { A ,  C }  =  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) ) )
65adantr 272 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) ) )
76imp 123 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  { A ,  C }  =  { B ,  C } )  -> 
( A  =  B  \/  A  =  C ) )
8 prid1g 3595 . . . . . . 7  |-  ( B  e.  _V  ->  B  e.  { B ,  C } )
9 eleq2 2179 . . . . . . 7  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  e. 
{ A ,  C } 
<->  B  e.  { B ,  C } ) )
108, 9syl5ibrcom 156 . . . . . 6  |-  ( B  e.  _V  ->  ( { A ,  C }  =  { B ,  C }  ->  B  e.  { A ,  C }
) )
11 elprg 3515 . . . . . 6  |-  ( B  e.  _V  ->  ( B  e.  { A ,  C }  <->  ( B  =  A  \/  B  =  C ) ) )
1210, 11sylibd 148 . . . . 5  |-  ( B  e.  _V  ->  ( { A ,  C }  =  { B ,  C }  ->  ( B  =  A  \/  B  =  C ) ) )
1312adantl 273 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  ( B  =  A  \/  B  =  C ) ) )
1413imp 123 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  { A ,  C }  =  { B ,  C } )  -> 
( B  =  A  \/  B  =  C ) )
15 eqcom 2117 . . 3  |-  ( A  =  B  <->  B  =  A )
16 eqeq2 2125 . . 3  |-  ( A  =  C  ->  ( B  =  A  <->  B  =  C ) )
177, 14, 15, 16oplem1 942 . 2  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  { A ,  C }  =  { B ,  C } )  ->  A  =  B )
1817ex 114 1  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  C }  =  { B ,  C }  ->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 680    = 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:  preqr2g  3662
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