Theorem disjssun 3514

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 3410	. . . . 5 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
2	1	equncomi 3309	. . . 4 |- ( A i^i ( B u. C ) ) = ( ( A i^i C ) u. ( A i^i B ) )
3		uneq2 3311	. . . . 5 |- ( ( A i^i B ) = (/) -> ( ( A i^i C ) u. ( A i^i B ) ) = ( ( A i^i C ) u. (/) ) )
4		un0 3484	. . . . 5 |- ( ( A i^i C ) u. (/) ) = ( A i^i C )
5	3, 4	eqtrdi 2245	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( A i^i C ) u. ( A i^i B ) ) = ( A i^i C ) )
6	2, 5	eqtrid 2241	. . 3 |- ( ( A i^i B ) = (/) -> ( A i^i ( B u. C ) ) = ( A i^i C ) )
7	6	eqeq1d 2205	. 2 |- ( ( A i^i B ) = (/) -> ( ( A i^i ( B u. C ) ) = A <-> ( A i^i C ) = A ) )
8		df-ss 3170	. 2 |- ( A C_ ( B u. C ) <-> ( A i^i ( B u. C ) ) = A )
9		df-ss 3170	. 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 1364   u. cun 3155   i^i cin 3156   C_ wss 3157   (/)c0 3450

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451

This theorem is referenced by: (None)