Theorem opnz 5457

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 4862	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
2	1	necon1ai 2985	. 2 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		dfopg 4837	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4		snex 5412	. . . . 5 ⊢ {𝐴} ∈ V
5	4	prnz 4744	. . . 4 ⊢ {{𝐴}, {𝐴, 𝐵}} ≠ ∅
6	5	a1i 11	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} ≠ ∅)
7	3, 6	eqnetrd 3025	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ≠ ∅)
8	2, 7	impbii 212	1 ⊢ (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4287  {csn 4590  {cpr 4592  ⟨cop 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597

This theorem is referenced by:  opnzi  5458  opeqex  5483  opelopabsb  5516  setsfun0  17233  fmlaomn0  35863