Theorem opelvvdif 38248

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 3924	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2		opelvvg 5679	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
3	2	biantrurd 532	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)))
4	1, 3	bitr4id 290	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Vcvv 3447   ∖ cdif 3911  ⟨cop 4595   × cxp 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5170  df-xp 5644

This theorem is referenced by:  vvdifopab  38249  brvvdif  38252