ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  preqr1 Unicode version

Theorem preqr1 3731
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 3665 . . . 4  |-  A  e. 
{ A ,  C }
3 eleq2 2221 . . . 4  |-  ( { A ,  C }  =  { B ,  C }  ->  ( A  e. 
{ A ,  C } 
<->  A  e.  { B ,  C } ) )
42, 3mpbii 147 . . 3  |-  ( { A ,  C }  =  { B ,  C }  ->  A  e.  { B ,  C }
)
51elpr 3581 . . 3  |-  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) )
64, 5sylib 121 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
7 preqr1.2 . . . . 5  |-  B  e. 
_V
87prid1 3665 . . . 4  |-  B  e. 
{ B ,  C }
9 eleq2 2221 . . . 4  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  e. 
{ A ,  C } 
<->  B  e.  { B ,  C } ) )
108, 9mpbiri 167 . . 3  |-  ( { A ,  C }  =  { B ,  C }  ->  B  e.  { A ,  C }
)
117elpr 3581 . . 3  |-  ( B  e.  { A ,  C }  <->  ( B  =  A  \/  B  =  C ) )
1210, 11sylib 121 . 2  |-  ( { A ,  C }  =  { B ,  C }  ->  ( B  =  A  \/  B  =  C ) )
13 eqcom 2159 . 2  |-  ( A  =  B  <->  B  =  A )
14 eqeq2 2167 . 2  |-  ( A  =  C  ->  ( B  =  A  <->  B  =  C ) )
156, 12, 13, 14oplem1 960 1  |-  ( { A ,  C }  =  { B ,  C }  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 698    = wceq 1335    e. wcel 2128   _Vcvv 2712   {cpr 3561
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567
This theorem is referenced by:  preqr2  3732
  Copyright terms: Public domain W3C validator