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Theorem opi1 4580
Description: One of the two elements in an ordered pair. (Contributed by NM, 15-Jul-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
opi1  |-  { A }  e.  <. A ,  B >.

Proof of Theorem opi1
StepHypRef Expression
1 snex 4554 . . 3  |-  { A }  e.  _V
21prid1 4004 . 2  |-  { A }  e.  { { A } ,  { A ,  B } }
3 opi1.1 . . 3  |-  A  e. 
_V
4 opi1.2 . . 3  |-  B  e. 
_V
53, 4dfop 4079 . 2  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
62, 5eleqtrri 2516 1  |-  { A }  e.  <. A ,  B >.
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1756   _Vcvv 2993   {csn 3898   {cpr 3900   <.cop 3904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905
This theorem is referenced by:  opth1  4586  opth  4587
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