Theorem opprc1 4866

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4865. (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 487	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2		opprc 4865	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
3	1, 2	nsyl5 160	1 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  ⟨cop 4600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-dif 3916  df-ss 3930  df-nul 4295  df-if 4493  df-op 4601

This theorem is referenced by:  snopeqop  5490  epelg  5563  brprcneu  6872  brprcneuALT  6873  fmlafvel  35776  bj-inftyexpidisj  37742  eu2ndop1stv  47751