Theorem opelvvdif 38863

Description: Negated elementhood of ordered pair. (Contributed by Peter Mazsa, 14-Jan-2019.)

Assertion

Ref	Expression
opelvvdif	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))

Proof of Theorem opelvvdif

Step	Hyp	Ref	Expression
1		eldif 3923	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2		opelvvg 5706	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
3	2	biantrurd 541	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)))
4	1, 3	bitr4id 293	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2150  Vcvv 3462   ∖ cdif 3910  ⟨cop 4600   × cxp 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5179  df-xp 5671

This theorem is referenced by:  vvdifopab  38864  brvvdif  38867