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Theorem opnz 4135
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 3717 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
21necon1ai 2454 . 2  |-  ( <. A ,  B >.  =/=  (/)  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 3694 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
4 snex 4110 . . . . 5  |-  { A }  e.  _V
54prnz 3649 . . . 4  |-  { { A } ,  { A ,  B } }  =/=  (/)
65a1i 12 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =/=  (/) )
73, 6eqnetrd 2430 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =/=  (/) )
82, 7impbii 182 1  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1621    =/= wne 2412   _Vcvv 2727   (/)c0 3362   {csn 3544   {cpr 3545   <.cop 3547
This theorem is referenced by:  opnzi  4136  opeqex  4150  opelopabsb  4168
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553
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