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Theorem undisj2 3982
 Description: The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
Assertion
Ref Expression
undisj2 (((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)

Proof of Theorem undisj2
StepHypRef Expression
1 un00 3963 . 2 (((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ ((𝐴𝐵) ∪ (𝐴𝐶)) = ∅)
2 indi 3832 . . 3 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
32eqeq1i 2615 . 2 ((𝐴 ∩ (𝐵𝐶)) = ∅ ↔ ((𝐴𝐵) ∪ (𝐴𝐶)) = ∅)
41, 3bitr4i 266 1 (((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∪ cun 3538   ∩ cin 3539  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875 This theorem is referenced by:  f1oun2prg  13512
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