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Theorem opnzi 4869
 Description: An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
opth1.1 𝐴 ∈ V
opth1.2 𝐵 ∈ V
Assertion
Ref Expression
opnzi 𝐴, 𝐵⟩ ≠ ∅

Proof of Theorem opnzi
StepHypRef Expression
1 opth1.1 . 2 𝐴 ∈ V
2 opth1.2 . 2 𝐵 ∈ V
3 opnz 4868 . 2 (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 957 1 𝐴, 𝐵⟩ ≠ ∅
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  ⟨cop 4131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132 This theorem is referenced by:  opelopabsb  4910  0nelxp  5067  0nelxpOLD  5068  unixp0  5586  funopsn  6319  0neqopab  6596  finxpreclem2  32403  finxp0  32404  finxpreclem6  32409
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