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Theorem opnz 4434
 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 4007 . . 3
21necon1ai 2648 . 2
3 dfopg 3984 . . 3
4 snex 4407 . . . . 5
54prnz 3925 . . . 4
65a1i 11 . . 3
73, 6eqnetrd 2621 . 2
82, 7impbii 182 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wcel 1726   wne 2601  cvv 2958  c0 3630  csn 3816  cpr 3817  cop 3819 This theorem is referenced by:  opnzi  4435  opeqex  4449  opelopabsb  4467 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825
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