MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opi1 Unicode version

Theorem opi1 4240
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 4216 . . 3  |-  { A }  e.  _V
21prid1 3734 . 2  |-  { A }  e.  { { A } ,  { A ,  B } }
3 opi1.1 . . 3  |-  A  e. 
_V
4 opi1.2 . . 3  |-  B  e. 
_V
53, 4dfop 3795 . 2  |-  <. A ,  B >.  =  { { A } ,  { A ,  B } }
62, 5eleqtrri 2356 1  |-  { A }  e.  <. A ,  B >.
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788   {csn 3640   {cpr 3641   <.cop 3643
This theorem is referenced by:  opth1  4244  opth  4245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
  Copyright terms: Public domain W3C validator