Theorem opprc2 4903

Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 4901. (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 484	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
2		opprc 4901	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
3	1, 2	nsyl5 159	1 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ∅c0 4339  ⟨cop 4637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-dif 3966  df-ss 3980  df-nul 4340  df-if 4532  df-op 4638

This theorem is referenced by:  snopeqop  5516  dmsnopss  6236  strle1  17192