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Theorem opi1 3995
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 opi1.1 . . . 4  |-  A  e. 
_V
21snex 3965 . . 3  |-  { A }  e.  _V
32prid1 3506 . 2  |-  { A }  e.  { { A } ,  { A ,  B } }
4 opi1.2 . . 3  |-  B  e. 
_V
51, 4dfop 3577 . 2  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
63, 5eleqtrri 2155 1  |-  { A }  e.  <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    e. wcel 1434   _Vcvv 2602   {csn 3406   {cpr 3407   <.cop 3409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415
This theorem is referenced by:  opth1  3999  opth  4000
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