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Theorem preqr1 4185
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 4119 . . . 4  |-  A  e. 
{ A ,  C }
3 eleq2 2514 . . . 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 4028 . . 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 4119 . . . 4  |-  B  e. 
{ B ,  C }
9 eleq2 2514 . . . 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 4028 . . 3  |-  ( B  e.  { A ,  C }  <->  ( B  =  A  \/  B  =  C ) )
1210, 11sylib 196 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  =  A  \/  B  =  C ) )
13 eqcom 2450 . 2  |-  ( A  =  B  <->  B  =  A )
14 eqeq2 2456 . 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 368    = wceq 1381    e. wcel 1802   _Vcvv 3093   {cpr 4012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-v 3095  df-un 3463  df-sn 4011  df-pr 4013
This theorem is referenced by:  preqr2  4186  opthwiener  4735  cusgrafilem2  24345  2pthfrgra  24876  wopprc  30940
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