Theorem disjssun 3574

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 3470	. . . . 5 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
2	1	equncomi 3367	. . . 4 |- ( A i^i ( B u. C ) ) = ( ( A i^i C ) u. ( A i^i B ) )
3		uneq2 3369	. . . . 5 |- ( ( A i^i B ) = (/) -> ( ( A i^i C ) u. ( A i^i B ) ) = ( ( A i^i C ) u. (/) ) )
4		un0 3544	. . . . 5 |- ( ( A i^i C ) u. (/) ) = ( A i^i C )
5	3, 4	eqtrdi 2283	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( A i^i C ) u. ( A i^i B ) ) = ( A i^i C ) )
6	2, 5	eqtrid 2279	. . 3 |- ( ( A i^i B ) = (/) -> ( A i^i ( B u. C ) ) = ( A i^i C ) )
7	6	eqeq1d 2243	. 2 |- ( ( A i^i B ) = (/) -> ( ( A i^i ( B u. C ) ) = A <-> ( A i^i C ) = A ) )
8		df-ss 3226	. 2 |- ( A C_ ( B u. C ) <-> ( A i^i ( B u. C ) ) = A )
9		df-ss 3226	. 2 |- ( A C_ C <-> ( A i^i C ) = A )
10	7, 8, 9	3bitr4g 223	1 |- ( ( A i^i B ) = (/) -> ( A C_ ( B u. C ) <-> A C_ C ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   u. cun 3211   i^i cin 3212   C_ wss 3213   (/)c0 3510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511

This theorem is referenced by:  hashfibclem  11210