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Theorem opnzi 5353
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 5352 . 2 (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 710 1 𝐴, 𝐵⟩ ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  wne 3014  Vcvv 3480  c0 4276  cop 4556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557
This theorem is referenced by:  opelopabsb  5404  0nelopab  5439  0nelxp  5576  unixp0  6121  funopsn  6901  cnfldfunALT  20160  fmlaomn0  32697  finxpreclem2  34755  finxp0  34756  finxpreclem6  34761
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