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Theorem opnzi 4435
Description: An ordered pair is nonempty if the arguments are sets. (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
StepHypRef Expression
1 opth1.1 . 2  |-  A  e. 
_V
2 opth1.2 . 2  |-  B  e. 
_V
3 opnz 4434 . 2  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
41, 2, 3mpbir2an 888 1  |-  <. A ,  B >.  =/=  (/)
Colors of variables: wff set class
Syntax hints:    e. wcel 1726    =/= wne 2601   _Vcvv 2958   (/)c0 3630   <.cop 3819
This theorem is referenced by:  opelopabsb  4467  0nelxp  4908  unixp0  5405  0neqopab  6121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825
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