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Theorem opnzi 4062
Description: An ordered pair is nonempty if the arguments are sets (it is also inhabited; see opm 4061). (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opth1.1  |-  A  e. 
_V
opth1.2  |-  B  e. 
_V
Assertion
Ref Expression
opnzi  |-  <. A ,  B >.  =/=  (/)

Proof of Theorem opnzi
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 opth1.1 . . 3  |-  A  e. 
_V
2 opth1.2 . . 3  |-  B  e. 
_V
3 opm 4061 . . 3  |-  ( E. x  x  e.  <. A ,  B >.  <->  ( A  e.  _V  /\  B  e. 
_V ) )
41, 2, 3mpbir2an 888 . 2  |-  E. x  x  e.  <. A ,  B >.
5 n0r 3296 . 2  |-  ( E. x  x  e.  <. A ,  B >.  ->  <. A ,  B >.  =/=  (/) )
64, 5ax-mp 7 1  |-  <. A ,  B >.  =/=  (/)
Colors of variables: wff set class
Syntax hints:   E.wex 1426    e. wcel 1438    =/= wne 2255   _Vcvv 2619   (/)c0 3286   <.cop 3449
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-in1 579  ax-in2 580  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 3957  ax-pow 4009
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455
This theorem is referenced by:  0nelxp  4465  0neqopab  5694
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