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Theorem elop 4058
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 3621 . . 3  |-  <. B ,  C >.  =  { { B } ,  { B ,  C } }
43eleq2i 2154 . 2  |-  ( A  e.  <. B ,  C >.  <-> 
A  e.  { { B } ,  { B ,  C } } )
5 elop.1 . . 3  |-  A  e. 
_V
65elpr 3467 . 2  |-  ( A  e.  { { B } ,  { B ,  C } }  <->  ( A  =  { B }  \/  A  =  { B ,  C } ) )
74, 6bitri 182 1  |-  ( A  e.  <. B ,  C >.  <-> 
( A  =  { B }  \/  A  =  { B ,  C } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3446   {cpr 3447   <.cop 3449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455
This theorem is referenced by:  relop  4586  bdop  11721
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