MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opprc1 Structured version   Visualization version   GIF version

Theorem opprc1 4834
Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4833. (Contributed by NM, 10-Apr-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opprc1 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Proof of Theorem opprc1
StepHypRef Expression
1 simpl 483 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2 opprc 4833 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
31, 2nsyl5 159 1 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1542  wcel 2110  Vcvv 3431  c0 4262  cop 4573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-dif 3895  df-nul 4263  df-if 4466  df-op 4574
This theorem is referenced by:  snopeqop  5424  epelg  5497  brprcneu  6761  brprcneuALT  6762  fmlafvel  33343  bj-inftyexpidisj  35377  eu2ndop1stv  44585
  Copyright terms: Public domain W3C validator