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Theorem nelun 30272
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 2904 . . . 4 (𝐴 = (𝐵𝐶) → (𝑋𝐴𝑋 ∈ (𝐵𝐶)))
2 elun 4109 . . . 4 (𝑋 ∈ (𝐵𝐶) ↔ (𝑋𝐵𝑋𝐶))
31, 2syl6bb 290 . . 3 (𝐴 = (𝐵𝐶) → (𝑋𝐴 ↔ (𝑋𝐵𝑋𝐶)))
43notbid 321 . 2 (𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ ¬ (𝑋𝐵𝑋𝐶)))
5 ioran 981 . 2 (¬ (𝑋𝐵𝑋𝐶) ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶))
64, 5syl6bb 290 1 (𝐴 = (𝐵𝐶) → (¬ 𝑋𝐴 ↔ (¬ 𝑋𝐵 ∧ ¬ 𝑋𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  cun 3916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3481  df-un 3923
This theorem is referenced by:  cycpmco2  30795
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