Theorem difdifdirss 3522

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 3413	. . . . 5 |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )
2		invdif 3392	. . . . 5 |- ( ( A \ C ) i^i ( _V \ B ) ) = ( ( A \ C ) \ B )
3	1, 2	eqtr4i 2213	. . . 4 |- ( ( A \ B ) \ C ) = ( ( A \ C ) i^i ( _V \ B ) )
4		un0 3471	. . . 4 |- ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) ) = ( ( A \ C ) i^i ( _V \ B ) )
5	3, 4	eqtr4i 2213	. . 3 |- ( ( A \ B ) \ C ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) )
6		indi 3397	. . . 4 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. ( ( A \ C ) i^i C ) )
7		disjdif 3510	. . . . . 6 |- ( C i^i ( A \ C ) ) = (/)
8		incom 3342	. . . . . 6 |- ( C i^i ( A \ C ) ) = ( ( A \ C ) i^i C )
9	7, 8	eqtr3i 2212	. . . . 5 |- (/) = ( ( A \ C ) i^i C )
10	9	uneq2i 3301	. . . 4 |- ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. ( ( A \ C ) i^i C ) )
11	6, 10	eqtr4i 2213	. . 3 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) )
12	5, 11	eqtr4i 2213	. 2 |- ( ( A \ B ) \ C ) = ( ( A \ C ) i^i ( ( _V \ B ) u. C ) )
13		ddifss 3388	. . . . . 6 |- C C_ ( _V \ ( _V \ C ) )
14		unss2 3321	. . . . . 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 3409	. . . . . 6 |- ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) ) C_ ( _V \ ( B i^i ( _V \ C ) ) )
17		invdif 3392	. . . . . . 7 |- ( B i^i ( _V \ C ) ) = ( B \ C )
18	17	difeq2i 3265	. . . . . 6 |- ( _V \ ( B i^i ( _V \ C ) ) ) = ( _V \ ( B \ C ) )
19	16, 18	sseqtri 3204	. . . . 5 |- ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) ) C_ ( _V \ ( B \ C ) )
20	15, 19	sstri 3179	. . . 4 |- ( ( _V \ B ) u. C ) C_ ( _V \ ( B \ C ) )
21		sslin 3376	. . . 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 3392	. . 3 |- ( ( A \ C ) i^i ( _V \ ( B \ C ) ) ) = ( ( A \ C ) \ ( B \ C ) )
24	22, 23	sseqtri 3204	. 2 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) C_ ( ( A \ C ) \ ( B \ C ) )
25	12, 24	eqsstri 3202	1 |- ( ( A \ B ) \ C ) C_ ( ( A \ C ) \ ( B \ C ) )

Syntax hints:   _Vcvv 2752   \ cdif 3141   u. cun 3142   i^i cin 3143   C_ wss 3144   (/)c0 3437

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438

This theorem is referenced by: (None)