Theorem opprc1 4857

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc 4856. (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 488	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
2		opprc 4856	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
3	1, 2	nsyl5 160	1 ⊢ (¬ 𝐴 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146  Vcvv 3453  ∅c0 4280  ⟨cop 4837

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ss 3900  df-dif 4049  df-nul 4281  df-if 4485  df-op 4838

This theorem is used by:  snopeqop  5481  epelg  5555  brprcneu  6883  brprcneuALT  6884  fmlafvel  36271  bj-inftyexpidisj  38271  eu2ndop1stv  48334