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Theorem opnz 4345
Description: An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opnz  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)

Proof of Theorem opnz
StepHypRef Expression
1 opprc 3919 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
21necon1ai 2571 . 2  |-  ( <. A ,  B >.  =/=  (/)  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 3896 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
4 snex 4318 . . . . 5  |-  { A }  e.  _V
54prnz 3838 . . . 4  |-  { { A } ,  { A ,  B } }  =/=  (/)
65a1i 10 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =/=  (/) )
73, 6eqnetrd 2547 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =/=  (/) )
82, 7impbii 180 1  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1715    =/= wne 2529   _Vcvv 2873   (/)c0 3543   {csn 3729   {cpr 3730   <.cop 3732
This theorem is referenced by:  opnzi  4346  opeqex  4360  opelopabsb  4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738
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