Theorem opnzi 5494

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 5493	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	1, 2, 3	mpbir2an 710	1 ⊢ ⟨𝐴, 𝐵⟩ ≠ ∅

Syntax hints:   ∈ wcel 2108   ≠ wne 2946  Vcvv 3488  ∅c0 4352  ⟨cop 4654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655

This theorem is referenced by:  opelopabsb  5549  0nelopab  5586  0nelopabOLD  5587  0nelxp  5734  unixp0  6314  funopsn  7182  cnfldfun  21401  cnfldfunOLD  21414  fmlaomn0  35358  finxpreclem2  37356  finxp0  37357  finxpreclem6  37362