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Theorem opnzi 5485
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 5484 . 2 (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
41, 2, 3mpbir2an 711 1 𝐴, 𝐵⟩ ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wne 2938  Vcvv 3478  c0 4339  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  opelopabsb  5540  0nelopab  5577  0nelxp  5723  unixp0  6305  funopsn  7168  cnfldfun  21396  cnfldfunOLD  21409  fmlaomn0  35375  finxpreclem2  37373  finxp0  37374  finxpreclem6  37379
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