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

Step	Hyp	Ref	Expression
1		indi 3743	. . . . 5
2	1	equncomi 3649	. . . 4
3		uneq2 3651	. . . . 5
4		un0 3810	. . . . 5
5	3, 4	syl6eq 2514	. . . 4
6	2, 5	syl5eq 2510	. . 3
7	6	eqeq1d 2459	. 2
8		df-ss 3489	. 2
9		df-ss 3489	. 2
10	7, 8, 9	3bitr4g 288	1

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