Theorem opnz 5477

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 4895	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
2	1	necon1ai 2967	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		dfopg 4870	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4		snex 5435	. . . . 5 ⊢ {𝐴} ∈ V
5	4	prnz 4776	. . . 4 ⊢ {{𝐴}, {𝐴, 𝐵}} ≠ ∅
6	5	a1i 11	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} ≠ ∅)
7	3, 6	eqnetrd 3007	. 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 2107   ≠ wne 2939  Vcvv 3479  ∅c0 4332  {csn 4625  {cpr 4627  ⟨cop 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632

This theorem is referenced by:  opnzi  5478  opeqex  5502  opelopabsb  5534  setsfun0  17210  fmlaomn0  35396