Theorem disjssun 3431

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 3328	. . . . 5 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
2	1	equncomi 3227	. . . 4 |- ( A i^i ( B u. C ) ) = ( ( A i^i C ) u. ( A i^i B ) )
3		uneq2 3229	. . . . 5 |- ( ( A i^i B ) = (/) -> ( ( A i^i C ) u. ( A i^i B ) ) = ( ( A i^i C ) u. (/) ) )
4		un0 3401	. . . . 5 |- ( ( A i^i C ) u. (/) ) = ( A i^i C )
5	3, 4	eqtrdi 2189	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( A i^i C ) u. ( A i^i B ) ) = ( A i^i C ) )
6	2, 5	syl5eq 2185	. . 3 |- ( ( A i^i B ) = (/) -> ( A i^i ( B u. C ) ) = ( A i^i C ) )
7	6	eqeq1d 2149	. 2 |- ( ( A i^i B ) = (/) -> ( ( A i^i ( B u. C ) ) = A <-> ( A i^i C ) = A ) )
8		df-ss 3089	. 2 |- ( A C_ ( B u. C ) <-> ( A i^i ( B u. C ) ) = A )
9		df-ss 3089	. 2 |- ( A C_ C <-> ( A i^i C ) = A )
10	7, 8, 9	3bitr4g 222	1 |- ( ( A i^i B ) = (/) -> ( A C_ ( B u. C ) <-> A C_ C ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   u. cun 3074   i^i cin 3075   C_ wss 3076   (/)c0 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369

This theorem is referenced by: (None)