MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opnz Unicode version

Theorem opnz 4392
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 3965 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
21necon1ai 2609 . 2  |-  ( <. A ,  B >.  =/=  (/)  ->  ( A  e. 
_V  /\  B  e.  _V ) )
3 dfopg 3942 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
4 snex 4365 . . . . 5  |-  { A }  e.  _V
54prnz 3883 . . . 4  |-  { { A } ,  { A ,  B } }  =/=  (/)
65a1i 11 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { { A } ,  { A ,  B } }  =/=  (/) )
73, 6eqnetrd 2585 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =/=  (/) )
82, 7impbii 181 1  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721    =/= wne 2567   _Vcvv 2916   (/)c0 3588   {csn 3774   {cpr 3775   <.cop 3777
This theorem is referenced by:  opnzi  4393  opeqex  4407  opelopabsb  4425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783
  Copyright terms: Public domain W3C validator