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Theorem undisj2 4469
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 4451 . 2 (((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ ((𝐴𝐵) ∪ (𝐴𝐶)) = ∅)
2 indi 4290 . . 3 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
32eqeq1i 2740 . 2 ((𝐴 ∩ (𝐵𝐶)) = ∅ ↔ ((𝐴𝐵) ∪ (𝐴𝐶)) = ∅)
41, 3bitr4i 278 1 (((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  cun 3961  cin 3962  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340
This theorem is referenced by:  disjtp2  4721  f1oun2prg  14953  cnfldfunALT  21397  cnfldfunALTOLD  21410  cnfldfunALTOLDOLD  21411
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