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Theorem bj-elopg 32845
Description: Characterization of the elements of an ordered pair (closed form of elop 4665, which actually has one unnecessary hypothesis). (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-elopg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  <. A ,  B >.  <->  ( C  =  { A }  \/  C  =  { A ,  B } ) ) )

Proof of Theorem bj-elopg
StepHypRef Expression
1 dfopg 4164 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
2 eleq2 2527 . . 3  |-  ( <. A ,  B >.  =  { { A } ,  { A ,  B } }  ->  ( C  e.  <. A ,  B >.  <-> 
C  e.  { { A } ,  { A ,  B } } ) )
3 snex 4640 . . . . 5  |-  { A }  e.  _V
4 prex 4641 . . . . 5  |-  { A ,  B }  e.  _V
53, 4elpr2 4003 . . . 4  |-  ( C  e.  { { A } ,  { A ,  B } }  <->  ( C  =  { A }  \/  C  =  { A ,  B } ) )
65a1i 11 . . 3  |-  ( <. A ,  B >.  =  { { A } ,  { A ,  B } }  ->  ( C  e.  { { A } ,  { A ,  B } }  <->  ( C  =  { A }  \/  C  =  { A ,  B } ) ) )
72, 6bitrd 253 . 2  |-  ( <. A ,  B >.  =  { { A } ,  { A ,  B } }  ->  ( C  e.  <. A ,  B >.  <-> 
( C  =  { A }  \/  C  =  { A ,  B } ) ) )
81, 7syl 16 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( C  e.  <. A ,  B >.  <->  ( C  =  { A }  \/  C  =  { A ,  B } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   {csn 3984   {cpr 3986   <.cop 3990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991
This theorem is referenced by:  bj-inftyexpidisj  32856
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