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Theorem opi2 4644
Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) (Avoid depending on this detail.)
Hypotheses
Ref Expression
opi1.1  |-  A  e. 
_V
opi1.2  |-  B  e. 
_V
Assertion
Ref Expression
opi2  |-  { A ,  B }  e.  <. A ,  B >.

Proof of Theorem opi2
StepHypRef Expression
1 prex 4618 . . 3  |-  { A ,  B }  e.  _V
21prid2 4068 . 2  |-  { A ,  B }  e.  { { A } ,  { A ,  B } }
3 opi1.1 . . 3  |-  A  e. 
_V
4 opi1.2 . . 3  |-  B  e. 
_V
53, 4dfop 4142 . 2  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
62, 5eleqtrri 2535 1  |-  { A ,  B }  e.  <. A ,  B >.
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1757   _Vcvv 3054   {csn 3961   {cpr 3963   <.cop 3967
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 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
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 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968
This theorem is referenced by:  opeluu  4645  uniopel  4679  elvvuni  4983
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