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

Theorem opprc1 4903
Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4902. (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 481 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2 opprc 4902 . 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 394   = wceq 1534  wcel 2099  Vcvv 3462  c0 4325  cop 4639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-dif 3950  df-ss 3964  df-nul 4326  df-if 4534  df-op 4640
This theorem is referenced by:  snopeqop  5512  epelg  5587  brprcneu  6891  brprcneuALT  6892  fmlafvel  35213  bj-inftyexpidisj  36917  eu2ndop1stv  46738
  Copyright terms: Public domain W3C validator