Theorem opprc1 4855

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4854. (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

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2		opprc 4854	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
3	1, 2	nsyl5 159	1 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  ⟨cop 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-dif 3906  df-ss 3920  df-nul 4288  df-if 4482  df-op 4589

This theorem is referenced by:  snopeqop  5462  epelg  5533  brprcneu  6832  brprcneuALT  6833  fmlafvel  35601  bj-inftyexpidisj  37465  eu2ndop1stv  47485