Theorem opprc2 4861

Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 4859. (Contributed by NM, 15-Nov-1994.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opprc2	⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Proof of Theorem opprc2

Step	Hyp	Ref	Expression
1		simpr 490	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
2		opprc 4859	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
3	1, 2	nsyl5 160	1 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ∅c0 4282  ⟨cop 4593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-dif 3905  df-ss 3919  df-nul 4283  df-if 4486  df-op 4594

This theorem is used by:  snopeqop  5487  dmsnopss  6214  strle1  17254  nowisdomv  30940