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Theorem elop 4197
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 3755 . . 3  |-  <. B ,  C >.  =  { { B } ,  { B ,  C } }
43eleq2i 2320 . 2  |-  ( A  e.  <. B ,  C >.  <-> 
A  e.  { { B } ,  { B ,  C } } )
5 elop.1 . . 3  |-  A  e. 
_V
65elpr 3618 . 2  |-  ( A  e.  { { B } ,  { B ,  C } }  <->  ( A  =  { B }  \/  A  =  { B ,  C } ) )
74, 6bitri 242 1  |-  ( A  e.  <. B ,  C >.  <-> 
( A  =  { B }  \/  A  =  { B ,  C } ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ wo 359    = wceq 1619    e. wcel 1621   _Vcvv 2757   {csn 3600   {cpr 3601   <.cop 3603
This theorem is referenced by:  relop  4808
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609
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