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Theorem opnz 4258
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 3833 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
21necon1ai 2501 . 2  |-  ( <. A ,  B >.  =/=  (/)  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 3810 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
4 snex 4232 . . . . 5  |-  { A }  e.  _V
54prnz 3758 . . . 4  |-  { { A } ,  { A ,  B } }  =/=  (/)
65a1i 10 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =/=  (/) )
73, 6eqnetrd 2477 . 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 1696    =/= wne 2459   _Vcvv 2801   (/)c0 3468   {csn 3653   {cpr 3654   <.cop 3656
This theorem is referenced by:  opnzi  4259  opeqex  4273  opelopabsb  4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662
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