Theorem difdifdirss 3509

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 3400	. . . . 5 |- ( ( A \ B ) \ C ) = ( ( A \ C ) \ B )
2		invdif 3379	. . . . 5 |- ( ( A \ C ) i^i ( _V \ B ) ) = ( ( A \ C ) \ B )
3	1, 2	eqtr4i 2201	. . . 4 |- ( ( A \ B ) \ C ) = ( ( A \ C ) i^i ( _V \ B ) )
4		un0 3458	. . . 4 |- ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) ) = ( ( A \ C ) i^i ( _V \ B ) )
5	3, 4	eqtr4i 2201	. . 3 |- ( ( A \ B ) \ C ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) )
6		indi 3384	. . . 4 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. ( ( A \ C ) i^i C ) )
7		disjdif 3497	. . . . . 6 |- ( C i^i ( A \ C ) ) = (/)
8		incom 3329	. . . . . 6 |- ( C i^i ( A \ C ) ) = ( ( A \ C ) i^i C )
9	7, 8	eqtr3i 2200	. . . . 5 |- (/) = ( ( A \ C ) i^i C )
10	9	uneq2i 3288	. . . 4 |- ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. ( ( A \ C ) i^i C ) )
11	6, 10	eqtr4i 2201	. . 3 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) = ( ( ( A \ C ) i^i ( _V \ B ) ) u. (/) )
12	5, 11	eqtr4i 2201	. 2 |- ( ( A \ B ) \ C ) = ( ( A \ C ) i^i ( ( _V \ B ) u. C ) )
13		ddifss 3375	. . . . . 6 |- C C_ ( _V \ ( _V \ C ) )
14		unss2 3308	. . . . . 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 3396	. . . . . 6 |- ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) ) C_ ( _V \ ( B i^i ( _V \ C ) ) )
17		invdif 3379	. . . . . . 7 |- ( B i^i ( _V \ C ) ) = ( B \ C )
18	17	difeq2i 3252	. . . . . 6 |- ( _V \ ( B i^i ( _V \ C ) ) ) = ( _V \ ( B \ C ) )
19	16, 18	sseqtri 3191	. . . . 5 |- ( ( _V \ B ) u. ( _V \ ( _V \ C ) ) ) C_ ( _V \ ( B \ C ) )
20	15, 19	sstri 3166	. . . 4 |- ( ( _V \ B ) u. C ) C_ ( _V \ ( B \ C ) )
21		sslin 3363	. . . 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 3379	. . 3 |- ( ( A \ C ) i^i ( _V \ ( B \ C ) ) ) = ( ( A \ C ) \ ( B \ C ) )
24	22, 23	sseqtri 3191	. 2 |- ( ( A \ C ) i^i ( ( _V \ B ) u. C ) ) C_ ( ( A \ C ) \ ( B \ C ) )
25	12, 24	eqsstri 3189	1 |- ( ( A \ B ) \ C ) C_ ( ( A \ C ) \ ( B \ C ) )

Syntax hints:   _Vcvv 2739   \ cdif 3128   u. cun 3129   i^i cin 3130   C_ wss 3131   (/)c0 3424

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425

This theorem is referenced by: (None)