Theorem opelvvdif 39387

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 4052	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
2		opelvvg 5696	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
3	2	biantrurd 542	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  Vcvv 3454   ∖ cdif 4048  ⟨cop 4838   × cxp 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661

This theorem is used by:  vvdifopab  39388  brvvdif  39391