Theorem opnzi 5430

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

Step	Hyp	Ref	Expression
1		opth1.1	. 2 ⊢ 𝐴 ∈ V
2		opth1.2	. 2 ⊢ 𝐵 ∈ V
3		opnz 5429	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	1, 2, 3	mpbir2an 712	1 ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅

Syntax hints:   ∈ wcel 2114   ≠ wne 2933  Vcvv 3442  ∅c0 4287  ⟨cop 4588

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589

This theorem is referenced by:  opelopabsb  5486  0nelopab  5521  0nelxp  5666  unixp0  6249  funopsn  7103  cnfldfun  21335  cnfldfunOLD  21348  fmlaomn0  35606  finxpreclem2  37645  finxp0  37646  finxpreclem6  37651