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Theorem undisj2 4174
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 4154 . 2 (((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ ((𝐴𝐵) ∪ (𝐴𝐶)) = ∅)
2 indi 4016 . . 3 (𝐴 ∩ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
32eqeq1i 2765 . 2 ((𝐴 ∩ (𝐵𝐶)) = ∅ ↔ ((𝐴𝐵) ∪ (𝐴𝐶)) = ∅)
41, 3bitr4i 267 1 (((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  cun 3713  cin 3714  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059
This theorem is referenced by:  disjtp2  4396  f1oun2prg  13862  cnfldfun  19960
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