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Theorem undisj1 4224
Description: The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
undisj1 (((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐵) ∩ 𝐶) = ∅)

Proof of Theorem undisj1
StepHypRef Expression
1 un00 4207 . 2 (((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐶) ∪ (𝐵𝐶)) = ∅)
2 indir 4076 . . 3 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
32eqeq1i 2804 . 2 (((𝐴𝐵) ∩ 𝐶) = ∅ ↔ ((𝐴𝐶) ∪ (𝐵𝐶)) = ∅)
41, 3bitr4i 270 1 (((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐵) ∩ 𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  cun 3767  cin 3768  c0 4115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116
This theorem is referenced by:  disjtpsn  4440  disjtp2  4441  funtp  6157  prinfzo0  12762  f1oun2prg  14002  cnfldfun  20080
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