Theorem disjssun 3609

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

Assertion

Ref	Expression
disjssun	|- ((A i^i B) = (/) -> (A C_ (B u. C) <-> A C_ C))

Proof of Theorem disjssun

Step	Hyp	Ref	Expression
1		indi 3502	. . . . 5 |- (A i^i (B u. C)) = ((A i^i B) u. (A i^i C))
2	1	equncomi 3411	. . . 4 |- (A i^i (B u. C)) = ((A i^i C) u. (A i^i B))
3		uneq2 3413	. . . . 5 |- ((A i^i B) = (/) -> ((A i^i C) u. (A i^i B)) = ((A i^i C) u. (/)))
4		un0 3576	. . . . 5 |- ((A i^i C) u. (/)) = (A i^i C)
5	3, 4	syl6eq 2401	. . . 4 |- ((A i^i B) = (/) -> ((A i^i C) u. (A i^i B)) = (A i^i C))
6	2, 5	syl5eq 2397	. . 3 |- ((A i^i B) = (/) -> (A i^i (B u. C)) = (A i^i C))
7	6	eqeq1d 2361	. 2 |- ((A i^i B) = (/) -> ((A i^i (B u. C)) = A <-> (A i^i C) = A))
8		df-ss 3260	. 2 |- (A C_ (B u. C) <-> (A i^i (B u. C)) = A)
9		df-ss 3260	. 2 |- (A C_ C <-> (A i^i C) = A)
10	7, 8, 9	3bitr4g 279	1 |- ((A i^i B) = (/) -> (A C_ (B u. C) <-> A C_ C))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   u. cun 3208   i^i cin 3209   C_ wss 3258  (/)c0 3551

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by: (None)