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

Theorem opnz 4387
 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

Proof of Theorem opnz
StepHypRef Expression
1 opprc 3961 . . 3
21necon1ai 2606 . 2
3 dfopg 3938 . . 3
4 snex 4360 . . . . 5
54prnz 3880 . . . 4
65a1i 11 . . 3
73, 6eqnetrd 2582 . 2
82, 7impbii 181 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wcel 1721   wne 2564  cvv 2913  c0 3585  csn 3771  cpr 3772  cop 3774 This theorem is referenced by:  opnzi  4388  opeqex  4402  opelopabsb  4420 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 2382  ax-sep 4285  ax-nul 4293  ax-pr 4358 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 2388  df-cleq 2394  df-clel 2397  df-nfc 2526  df-ne 2566  df-v 2915  df-dif 3280  df-un 3282  df-in 3284  df-ss 3291  df-nul 3586  df-if 3697  df-sn 3777  df-pr 3778  df-op 3780
 Copyright terms: Public domain W3C validator