MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  preqr1 Structured version   Unicode version

Theorem preqr1 4195
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 4130 . . . 4  |-  A  e. 
{ A ,  C }
3 eleq2 2535 . . . 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 4040 . . 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 4130 . . . 4  |-  B  e. 
{ B ,  C }
9 eleq2 2535 . . . 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 4040 . . 3  |-  ( B  e.  { A ,  C }  <->  ( B  =  A  \/  B  =  C ) )
1210, 11sylib 196 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  =  A  \/  B  =  C ) )
13 eqcom 2471 . 2  |-  ( A  =  B  <->  B  =  A )
14 eqeq2 2477 . 2  |-  ( A  =  C  ->  ( B  =  A  <->  B  =  C ) )
156, 12, 13, 14oplem1 957 1  |-  ( { A ,  C }  =  { B ,  C }  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1374    e. wcel 1762   _Vcvv 3108   {cpr 4024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-v 3110  df-un 3476  df-sn 4023  df-pr 4025
This theorem is referenced by:  preqr2  4196  opthwiener  4744  cusgrafilem2  24144  2pthfrgra  24675  wopprc  30567
  Copyright terms: Public domain W3C validator