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Theorem opi1 4459
Description: One of the two elements in an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opi1.1  |-  A  e. 
_V
opi1.2  |-  B  e. 
_V
Assertion
Ref Expression
opi1  |-  { A }  e.  <. A ,  B >.

Proof of Theorem opi1
StepHypRef Expression
1 snex 4434 . . 3  |-  { A }  e.  _V
21prid1 3936 . 2  |-  { A }  e.  { { A } ,  { A ,  B } }
3 opi1.1 . . 3  |-  A  e. 
_V
4 opi1.2 . . 3  |-  B  e. 
_V
53, 4dfop 4007 . 2  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
62, 5eleqtrri 2515 1  |-  { A }  e.  <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    e. wcel 1727   _Vcvv 2962   {csn 3838   {cpr 3839   <.cop 3841
This theorem is referenced by:  opth1  4463  opth  4464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847
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