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Theorem preqr1 4118
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 4052 . . . 4  |-  A  e. 
{ A ,  C }
3 eleq2 2455 . . . 4  |-  ( { A ,  C }  =  { B ,  C }  ->  ( A  e. 
{ A ,  C } 
<->  A  e.  { B ,  C } ) )
42, 3mpbii 211 . . 3  |-  ( { A ,  C }  =  { B ,  C }  ->  A  e.  { B ,  C }
)
51elpr 3962 . . 3  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )
64, 5sylib 196 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
7 preqr1.2 . . . . 5  |-  B  e. 
_V
87prid1 4052 . . . 4  |-  B  e. 
{ B ,  C }
9 eleq2 2455 . . . 4  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  e. 
{ A ,  C } 
<->  B  e.  { B ,  C } ) )
108, 9mpbiri 233 . . 3  |-  ( { A ,  C }  =  { B ,  C }  ->  B  e.  { A ,  C }
)
117elpr 3962 . . 3  |-  ( B  e.  { A ,  C }  <->  ( B  =  A  \/  B  =  C ) )
1210, 11sylib 196 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  =  A  \/  B  =  C ) )
13 eqcom 2391 . 2  |-  ( A  =  B  <->  B  =  A )
14 eqeq2 2397 . 2  |-  ( A  =  C  ->  ( B  =  A  <->  B  =  C ) )
156, 12, 13, 14oplem1 962 1  |-  ( { A ,  C }  =  { B ,  C }  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    = wceq 1399    e. wcel 1826   _Vcvv 3034   {cpr 3946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-un 3394  df-sn 3945  df-pr 3947
This theorem is referenced by:  preqr2  4119  opthwiener  4663  cusgrafilem2  24601  2pthfrgra  25132  wopprc  31138
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