Theorem opprc2 4822

Description: Expansion of an ordered pair when the second member is a proper class. See also opprc 4820. (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 487	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
2	1	con3i 157	. 2 ⊢ (¬ 𝐵 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		opprc 4820	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
4	2, 3	syl 17	1 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3495  ∅c0 4291  ⟨cop 4567

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-op 4568

This theorem is referenced by:  snopeqop  5389  dmsnopss  6066  strle1  16586