Theorem opprc 4889

Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

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

Proof of Theorem opprc

Step	Hyp	Ref	Expression
1		dfopif 4863	. 2 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅)
2		iffalse 4530	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) = ∅)
3	1, 2	eqtrid 2776	1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466  ∅c0 4315  ifcif 4521  {csn 4621  {cpr 4623  ⟨cop 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3944  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-op 4628

This theorem is referenced by:  opprc1  4890  opprc2  4891  oprcl  4892  opnz  5464  brabv  5560  opswap  6219  brtpos  8216  bj-brrelex12ALT  36439  elnonrel  42850