Theorem nelun 32991

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 2849	. . . 4 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ (𝐵 ∪ 𝐶)))
2		elun 4100	. . . 4 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))
3	1, 2	bitrdi 290	. . 3 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)))
4	3	notbid 321	. 2 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ ¬ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶)))
5		ioran 999	. 2 ⊢ (¬ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶) ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶))
6	4, 5	bitrdi 290	1 ⊢ (𝐴 = (𝐵 ∪ 𝐶) → (¬ 𝑋 ∈ 𝐴 ↔ (¬ 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ 𝐶)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ∪ cun 3897

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904

This theorem is used by:  cycpmco2  33576  elrgspnlem4  33688  rprmnz  33933  rprmnunit  33934  rsprprmprmidlb  33936  rprmirredb  33945