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Theorem undisj1 3602
 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 3586 . 2 (((AC) = (BC) = ) ↔ ((AC) ∪ (BC)) = )
2 indir 3503 . . 3 ((AB) ∩ C) = ((AC) ∪ (BC))
32eqeq1i 2360 . 2 (((AB) ∩ C) = ↔ ((AC) ∪ (BC)) = )
41, 3bitr4i 243 1 (((AC) = (BC) = ) ↔ ((AB) ∩ C) = )
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   = wceq 1642   ∪ cun 3207   ∩ cin 3208  ∅c0 3550 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by: (None)
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