Theorem opelvvdif 38467

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 3912	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2		opelvvg 5666	. . 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 2114  Vcvv 3441   ∖ cdif 3899  ⟨cop 4587   × cxp 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-opab 5162  df-xp 5631

This theorem is referenced by:  vvdifopab  38468  brvvdif  38471