Theorem difdifdirss 3536

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 3427	. . . . 5 |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )
2		invdif 3406	. . . . 5 |- ( ( A \ C ) i^i ( _V \ B ) ) = ( ( A \ C ) \ B )
3	1, 2	eqtr4i 2220	. . . 4 |- ( ( A \ B ) \ C ) = ( ( A \ C ) i^i ( _V \ B ) )
4		un0 3485	. . . 4 |- ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) ) = ( ( A \ C ) i^i ( _V \ B ) )
5	3, 4	eqtr4i 2220	. . 3 |- ( ( A \ B ) \ C ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) )
6		indi 3411	. . . 4 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. ( ( A \ C ) i^i C ) )
7		disjdif 3524	. . . . . 6 |- ( C i^i ( A \ C ) ) = (/)
8		incom 3356	. . . . . 6 |- ( C i^i ( A \ C ) ) = ( ( A \ C ) i^i C )
9	7, 8	eqtr3i 2219	. . . . 5 |- (/) = ( ( A \ C ) i^i C )
10	9	uneq2i 3315	. . . 4 |- ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. ( ( A \ C ) i^i C ) )
11	6, 10	eqtr4i 2220	. . 3 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) )
12	5, 11	eqtr4i 2220	. 2 |- ( ( A \ B ) \ C ) = ( ( A \ C ) i^i ( ( _V \ B ) u. C ) )
13		ddifss 3402	. . . . . 6 |- C C_ ( _V \ ( _V \ C ) )
14		unss2 3335	. . . . . 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 3423	. . . . . 6 |- ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) ) C_ ( _V \ ( B i^i ( _V \ C ) ) )
17		invdif 3406	. . . . . . 7 |- ( B i^i ( _V \ C ) ) = ( B \ C )
18	17	difeq2i 3279	. . . . . 6 |- ( _V \ ( B i^i ( _V \ C ) ) ) = ( _V \ ( B \ C ) )
19	16, 18	sseqtri 3218	. . . . 5 |- ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) ) C_ ( _V \ ( B \ C ) )
20	15, 19	sstri 3193	. . . 4 |- ( ( _V \ B ) u. C ) C_ ( _V \ ( B \ C ) )
21		sslin 3390	. . . 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 3406	. . 3 |- ( ( A \ C ) i^i ( _V \ ( B \ C ) ) ) = ( ( A \ C ) \ ( B \ C ) )
24	22, 23	sseqtri 3218	. 2 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) C_ ( ( A \ C ) \ ( B \ C ) )
25	12, 24	eqsstri 3216	1 |- ( ( A \ B ) \ C ) C_ ( ( A \ C ) \ ( B \ C ) )

Syntax hints:   _Vcvv 2763   \ cdif 3154   u. cun 3155   i^i cin 3156   C_ wss 3157   (/)c0 3451

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-ral 2480  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452

This theorem is referenced by: (None)