MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opnz Structured version   Visualization version   GIF version

Theorem opnz 5464
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
StepHypRef Expression
1 opprc 4889 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
21necon1ai 2960 . 2 (⟨𝐴, 𝐵⟩ ≠ ∅ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 dfopg 4864 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4 snex 5422 . . . . 5 {𝐴} ∈ V
54prnz 4774 . . . 4 {{𝐴}, {𝐴, 𝐵}} ≠ ∅
65a1i 11 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {{𝐴}, {𝐴, 𝐵}} ≠ ∅)
73, 6eqnetrd 3000 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ ≠ ∅)
82, 7impbii 208 1 (⟨𝐴, 𝐵⟩ ≠ ∅ ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2098  wne 2932  Vcvv 3466  c0 4315  {csn 4621  {cpr 4623  cop 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628
This theorem is referenced by:  opnzi  5465  opeqex  5489  opelopabsb  5521  setsfun0  17106  fmlaomn0  34872
  Copyright terms: Public domain W3C validator