Theorem difdifdirss 3442

Description: Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
difdifdirss	|- ( ( A \ B ) \ C ) C_ ( ( A \ C ) \ ( B \ C ) )

Proof of Theorem difdifdirss

Step	Hyp	Ref	Expression
1		dif32 3334	. . . . 5 |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )
2		invdif 3313	. . . . 5 |- ( ( A \ C ) i^i ( _V \ B ) ) = ( ( A \ C ) \ B )
3	1, 2	eqtr4i 2161	. . . 4 |- ( ( A \ B ) \ C ) = ( ( A \ C ) i^i ( _V \ B ) )
4		un0 3391	. . . 4 |- ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) ) = ( ( A \ C ) i^i ( _V \ B ) )
5	3, 4	eqtr4i 2161	. . 3 |- ( ( A \ B ) \ C ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) )
6		indi 3318	. . . 4 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. ( ( A \ C ) i^i C ) )
7		disjdif 3430	. . . . . 6 |- ( C i^i ( A \ C ) ) = (/)
8		incom 3263	. . . . . 6 |- ( C i^i ( A \ C ) ) = ( ( A \ C ) i^i C )
9	7, 8	eqtr3i 2160	. . . . 5 |- (/) = ( ( A \ C ) i^i C )
10	9	uneq2i 3222	. . . 4 |- ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. ( ( A \ C ) i^i C ) )
11	6, 10	eqtr4i 2161	. . 3 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) )
12	5, 11	eqtr4i 2161	. 2 |- ( ( A \ B ) \ C ) = ( ( A \ C ) i^i ( ( _V \ B ) u. C ) )
13		ddifss 3309	. . . . . 6 |- C C_ ( _V \ ( _V \ C ) )
14		unss2 3242	. . . . . 6 |- ( C C_ ( _V \ ( _V \ C ) ) -> ( ( _V \ B ) u. C ) C_ ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) ) )
15	13, 14	ax-mp 5	. . . . 5 |- ( ( _V \ B ) u. C ) C_ ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) )
16		indmss 3330	. . . . . 6 |- ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) ) C_ ( _V \ ( B i^i ( _V \ C ) ) )
17		invdif 3313	. . . . . . 7 |- ( B i^i ( _V \ C ) ) = ( B \ C )
18	17	difeq2i 3186	. . . . . 6 |- ( _V \ ( B i^i ( _V \ C ) ) ) = ( _V \ ( B \ C ) )
19	16, 18	sseqtri 3126	. . . . 5 |- ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) ) C_ ( _V \ ( B \ C ) )
20	15, 19	sstri 3101	. . . 4 |- ( ( _V \ B ) u. C ) C_ ( _V \ ( B \ C ) )
21		sslin 3297	. . . 4 |- ( ( ( _V \ B ) u. C ) C_ ( _V \ ( B \ C ) ) -> ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) C_ ( ( A \ C ) i^i ( _V \ ( B \ C ) ) ) )
22	20, 21	ax-mp 5	. . 3 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) C_ ( ( A \ C ) i^i ( _V \ ( B \ C ) ) )
23		invdif 3313	. . 3 |- ( ( A \ C ) i^i ( _V \ ( B \ C ) ) ) = ( ( A \ C ) \ ( B \ C ) )
24	22, 23	sseqtri 3126	. 2 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) C_ ( ( A \ C ) \ ( B \ C ) )
25	12, 24	eqsstri 3124	1 |- ( ( A \ B ) \ C ) C_ ( ( A \ C ) \ ( B \ C ) )

Syntax hints:   _Vcvv 2681   \ cdif 3063   u. cun 3064   i^i cin 3065   C_ wss 3066   (/)c0 3358

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359

This theorem is referenced by: (None)