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Theorem opnzi 4728
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 4727 . 2  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
41, 2, 3mpbir2an 920 1  |-  <. A ,  B >.  =/=  (/)
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1819    =/= wne 2652   _Vcvv 3109   (/)c0 3793   <.cop 4038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039
This theorem is referenced by:  opelopabsb  4766  0nelxp  5036  unixp0  5547  0neqopab  6340
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