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Theorem elop 4153
Description: An ordered pair has two elements. Exercise 3 of [TakeutiZaring] p. 15. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
elop.1  |-  A  e. 
_V
elop.2  |-  B  e. 
_V
elop.3  |-  C  e. 
_V
Assertion
Ref Expression
elop  |-  ( A  e.  <. B ,  C >.  <-> 
( A  =  { B }  \/  A  =  { B ,  C } ) )

Proof of Theorem elop
StepHypRef Expression
1 elop.2 . . . 4  |-  B  e. 
_V
2 elop.3 . . . 4  |-  C  e. 
_V
31, 2dfop 3704 . . 3  |-  <. B ,  C >.  =  { { B } ,  { B ,  C } }
43eleq2i 2206 . 2  |-  ( A  e.  <. B ,  C >.  <-> 
A  e.  { { B } ,  { B ,  C } } )
5 elop.1 . . 3  |-  A  e. 
_V
65elpr 3548 . 2  |-  ( A  e.  { { B } ,  { B ,  C } }  <->  ( A  =  { B }  \/  A  =  { B ,  C } ) )
74, 6bitri 183 1  |-  ( A  e.  <. B ,  C >.  <-> 
( A  =  { B }  \/  A  =  { B ,  C } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   _Vcvv 2686   {csn 3527   {cpr 3528   <.cop 3530
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536
This theorem is referenced by:  relop  4689  bdop  13157
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