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Theorem elop 4255
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 3811 . . 3  |-  <. B ,  C >.  =  { { B } ,  { B ,  C } }
43eleq2i 2360 . 2  |-  ( A  e.  <. B ,  C >.  <-> 
A  e.  { { B } ,  { B ,  C } } )
5 elop.1 . . 3  |-  A  e. 
_V
65elpr 3671 . 2  |-  ( A  e.  { { B } ,  { B ,  C } }  <->  ( A  =  { B }  \/  A  =  { B ,  C } ) )
74, 6bitri 240 1  |-  ( A  e.  <. B ,  C >.  <-> 
( A  =  { B }  \/  A  =  { B ,  C } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   {cpr 3654   <.cop 3656
This theorem is referenced by:  relop  4850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662
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