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Theorem undisj1 4459
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 4440 . 2 (((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐶) ∪ (𝐵𝐶)) = ∅)
2 indir 4273 . . 3 ((𝐴𝐵) ∩ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
32eqeq1i 2738 . 2 (((𝐴𝐵) ∩ 𝐶) = ∅ ↔ ((𝐴𝐶) ∪ (𝐵𝐶)) = ∅)
41, 3bitr4i 278 1 (((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐵) ∩ 𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  cun 3944  cin 3945  c0 4320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321
This theorem is referenced by:  disjtpsn  4715  disjtp2  4716  funtp  6597  prinfzo0  13658  f1oun2prg  14855  cnfldfunALT  20931  cnfldfunALTOLD  20932
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