New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  undisj2 GIF version

Theorem undisj2 3603
 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 3586 . 2 (((AB) = (AC) = ) ↔ ((AB) ∪ (AC)) = )
2 indi 3501 . . 3 (A ∩ (BC)) = ((AB) ∪ (AC))
32eqeq1i 2360 . 2 ((A ∩ (BC)) = ↔ ((AB) ∪ (AC)) = )
41, 3bitr4i 243 1 (((AB) = (AC) = ) ↔ (A ∩ (BC)) = )
 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-mp 5  ax-1 6  ax-2 7  ax-3 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)
 Copyright terms: Public domain W3C validator