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Theorem nelun 32322
Description: Negated membership for a union. (Contributed by Thierry Arnoux, 13-Dec-2023.)
Assertion
Ref Expression
nelun (𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶)))

Proof of Theorem nelun
StepHypRef Expression
1 eleq2 2818 . . . 4 (𝐴 = (𝐵𝐶) → (𝑋𝐴𝑋 ∈ (𝐵𝐶)))
2 elun 4147 . . . 4 (𝑋 ∈ (𝐵𝐶) ↔ (𝑋𝐵𝑋𝐶))
31, 2bitrdi 287 . . 3 (𝐴 = (𝐵𝐶) → (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶)))
43notbid 318 . 2 (𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ ¬ (𝑋𝐵𝑋𝐶)))
5 ioran 982 . 2 (¬ (𝑋𝐵𝑋𝐶) ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶))
64, 5bitrdi 287 1 (𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846   = wceq 1534  wcel 2099  cun 3945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-un 3952
This theorem is referenced by:  cycpmco2  32867
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