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Theorem bj-elopg 35003
Description: Characterization of the elements of an ordered pair (closed form of elop 4703, 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 4201 . 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 4678 . . . 4  |-  { A }  e.  _V
4 prex 4679 . . . 4  |-  { A ,  B }  e.  _V
53, 4elpr2 4035 . . 3  |-  ( C  e.  { { A } ,  { A ,  B } }  <->  ( C  =  { A }  \/  C  =  { A ,  B } ) )
62, 5syl6bb 261 . 2  |-  ( <. A ,  B >.  =  { { A } ,  { A ,  B } }  ->  ( C  e.  <. A ,  B >.  <-> 
( C  =  { A }  \/  C  =  { A ,  B } ) ) )
71, 6syl 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   {csn 4016   {cpr 4018   <.cop 4022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023
This theorem is referenced by:  bj-inftyexpidisj  35013
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