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Theorem opi2 4051
Description: One of the two elements of 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
opi2  |-  { A ,  B }  e.  <. A ,  B >.

Proof of Theorem opi2
StepHypRef Expression
1 opi1.1 . . . 4  |-  A  e. 
_V
2 opi1.2 . . . 4  |-  B  e. 
_V
3 prexg 4029 . . . 4  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { A ,  B }  e.  _V )
41, 2, 3mp2an 417 . . 3  |-  { A ,  B }  e.  _V
54prid2 3544 . 2  |-  { A ,  B }  e.  { { A } ,  { A ,  B } }
61, 2dfop 3616 . 2  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
75, 6eleqtrri 2163 1  |-  { A ,  B }  e.  <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    e. wcel 1438   _Vcvv 2619   {csn 3441   {cpr 3442   <.cop 3444
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450
This theorem is referenced by:  uniopel  4074  opeluu  4263  elvvuni  4490
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