Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  disjssun Unicode version

Theorem disjssun 3884
 Description: Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
disjssun

Proof of Theorem disjssun
StepHypRef Expression
1 indi 3743 . . . . 5
21equncomi 3649 . . . 4
3 uneq2 3651 . . . . 5
4 un0 3810 . . . . 5
53, 4syl6eq 2514 . . . 4
62, 5syl5eq 2510 . . 3
76eqeq1d 2459 . 2
8 df-ss 3489 . 2
9 df-ss 3489 . 2
107, 8, 93bitr4g 288 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  u.cun 3473  i^icin 3474  C_wss 3475   c0 3784 This theorem is referenced by:  hashbclem  12501  alexsubALTlem2  20548  iccntr  21326  reconnlem1  21331  dvne0  22412 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785
 Copyright terms: Public domain W3C validator