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Theorem preqr1 3964
Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. (Contributed by NM, 18-Oct-1995.)
Hypotheses
Ref Expression
preqr1.1  |-  A  e. 
_V
preqr1.2  |-  B  e. 
_V
Assertion
Ref Expression
preqr1  |-  ( { A ,  C }  =  { B ,  C }  ->  A  =  B )

Proof of Theorem preqr1
StepHypRef Expression
1 preqr1.1 . . . . 5  |-  A  e. 
_V
21prid1 3904 . . . 4  |-  A  e. 
{ A ,  C }
3 eleq2 2496 . . . 4  |-  ( { A ,  C }  =  { B ,  C }  ->  ( A  e. 
{ A ,  C } 
<->  A  e.  { B ,  C } ) )
42, 3mpbii 203 . . 3  |-  ( { A ,  C }  =  { B ,  C }  ->  A  e.  { B ,  C }
)
51elpr 3824 . . 3  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )
64, 5sylib 189 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
7 preqr1.2 . . . . 5  |-  B  e. 
_V
87prid1 3904 . . . 4  |-  B  e. 
{ B ,  C }
9 eleq2 2496 . . . 4  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  e. 
{ A ,  C } 
<->  B  e.  { B ,  C } ) )
108, 9mpbiri 225 . . 3  |-  ( { A ,  C }  =  { B ,  C }  ->  B  e.  { A ,  C }
)
117elpr 3824 . . 3  |-  ( B  e.  { A ,  C }  <->  ( B  =  A  \/  B  =  C ) )
1210, 11sylib 189 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  =  A  \/  B  =  C ) )
13 eqcom 2437 . 2  |-  ( A  =  B  <->  B  =  A )
14 eqeq2 2444 . 2  |-  ( A  =  C  ->  ( B  =  A  <->  B  =  C ) )
156, 12, 13, 14oplem1 931 1  |-  ( { A ,  C }  =  { B ,  C }  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1652    e. wcel 1725   _Vcvv 2948   {cpr 3807
This theorem is referenced by:  preqr2  3965  opthwiener  4450  cusgrafilem2  21481  wopprc  27092  2pthfrgra  28338
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
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