Theorem nelun 32440

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 2823	. . . 4 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ (𝐵 ∪ 𝐶)))
2		elun 4128	. . . 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 985	. 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 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108   ∪ cun 3924

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931

This theorem is referenced by:  cycpmco2  33090  elrgspnlem4  33186  rprmnz  33481  rprmnunit  33482  rsprprmprmidlb  33484  rprmirredb  33493