Theorem opelvvdif 35514

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		opelvvg 5589	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ (V × V))
2	1	biantrurd 535	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅)))
3		eldif 3945	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))
4	2, 3	syl6rbbr 292	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ ((V × V) ∖ 𝑅) ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  Vcvv 3494   ∖ cdif 3932  ⟨cop 4566   × cxp 5547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-opab 5121  df-xp 5555

This theorem is referenced by:  vvdifopab  35515  brvvdif  35518