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Theorem opnzi 4687
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 4686 . 2  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
41, 2, 3mpbir2an 936 1  |-  <. A ,  B >.  =/=  (/)
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1897    =/= wne 2632   _Vcvv 3056   (/)c0 3742   <.cop 3985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986
This theorem is referenced by:  opelopabsb  4724  0nelxp  4880  unixp0  5388  0neqopab  6360  finxpreclem2  31826  finxp0  31827  finxpreclem6  31832  funopsn  39059
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