Theorem opnz 5426

Description: An ordered pair is nonempty iff the arguments are sets. (Contributed by NM, 24-Jan-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opnz	⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem opnz

Step	Hyp	Ref	Expression
1		opprc 4839	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
2	1	necon1ai 2959	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		dfopg 4814	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4		snex 5381	. . . . 5 ⊢ {𝐴} ∈ V
5	4	prnz 4721	. . . 4 ⊢ {{𝐴}, {𝐴, 𝐵}} ≠ ∅
6	5	a1i 11	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} ≠ ∅)
7	3, 6	eqnetrd 2999	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ≠ ∅)
8	2, 7	impbii 209	1 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   ≠ wne 2932  Vcvv 3429  ∅c0 4273  {csn 4567  {cpr 4569  ⟨cop 4573

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 2708  ax-sep 5231  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574

This theorem is referenced by:  opnzi  5427  opeqex  5452  opelopabsb  5485  setsfun0  17142  fmlaomn0  35572