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Theorem opnz 4244
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 3819 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
21necon1ai 2490 . 2  |-  ( <. A ,  B >.  =/=  (/)  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 3796 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
4 snex 4218 . . . . 5  |-  { A }  e.  _V
54prnz 3747 . . . 4  |-  { { A } ,  { A ,  B } }  =/=  (/)
65a1i 10 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =/=  (/) )
73, 6eqnetrd 2466 . 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 1686    =/= wne 2448   _Vcvv 2790   (/)c0 3457   {csn 3642   {cpr 3643   <.cop 3645
This theorem is referenced by:  opnzi  4245  opeqex  4259  opelopabsb  4277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651
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