Theorem disjssun 3500

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 3396	. . . . 5 |- ( A i^i ( B u. C ) ) = ( ( A i^i B ) u. ( A i^i C ) )
2	1	equncomi 3295	. . . 4 |- ( A i^i ( B u. C ) ) = ( ( A i^i C ) u. ( A i^i B ) )
3		uneq2 3297	. . . . 5 |- ( ( A i^i B ) = (/) -> ( ( A i^i C ) u. ( A i^i B ) ) = ( ( A i^i C ) u. (/) ) )
4		un0 3470	. . . . 5 |- ( ( A i^i C ) u. (/) ) = ( A i^i C )
5	3, 4	eqtrdi 2237	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( A i^i C ) u. ( A i^i B ) ) = ( A i^i C ) )
6	2, 5	eqtrid 2233	. . 3 |- ( ( A i^i B ) = (/) -> ( A i^i ( B u. C ) ) = ( A i^i C ) )
7	6	eqeq1d 2197	. 2 |- ( ( A i^i B ) = (/) -> ( ( A i^i ( B u. C ) ) = A <-> ( A i^i C ) = A ) )
8		df-ss 3156	. 2 |- ( A C_ ( B u. C ) <-> ( A i^i ( B u. C ) ) = A )
9		df-ss 3156	. 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 1363   u. cun 3141   i^i cin 3142   C_ wss 3143   (/)c0 3436

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437

This theorem is referenced by: (None)