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Theorem undisj1 4372
 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 4353 . 2 (((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐶) ∪ (𝐵𝐶)) = ∅)
2 indir 4205 . . 3 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
32eqeq1i 2803 . 2 (((𝐴𝐵) ∩ 𝐶) = ∅ ↔ ((𝐴𝐶) ∪ (𝐵𝐶)) = ∅)
41, 3bitr4i 281 1 (((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐵) ∩ 𝐶) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∪ cun 3881   ∩ cin 3882  ∅c0 4246 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247 This theorem is referenced by:  disjtpsn  4614  disjtp2  4615  funtp  6389  prinfzo0  13091  f1oun2prg  14290  cnfldfun  20124
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