Theorem nelun 30859

Description: Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)

Assertion

Ref	Expression
nelun	⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶)))

Proof of Theorem nelun

Step	Hyp	Ref	Expression
1		eleq2 2827	. . . 4 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ (𝐵 ∪ 𝐶)))
2		elun 4083	. . . 4 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))
3	1, 2	bitrdi 287	. . 3 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)))
4	3	notbid 318	. 2 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ ¬ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)))
5		ioran 981	. 2 ⊢ (¬ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶))
6	4, 5	bitrdi 287	1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ∪ cun 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892

This theorem is referenced by:  cycpmco2  31400