Theorem opnzi 5417

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 5416	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	1, 2, 3	mpbir2an 711	1 ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅

Syntax hints:   ∈ wcel 2109   ≠ wne 2925  Vcvv 3436  ∅c0 4284  ⟨cop 4583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584

This theorem is referenced by:  opelopabsb  5473  0nelopab  5508  0nelxp  5653  unixp0  6231  funopsn  7082  cnfldfun  21275  cnfldfunOLD  21288  fmlaomn0  35367  finxpreclem2  37368  finxp0  37369  finxpreclem6  37374