Theorem difdif2ss 3464

Description: Set difference with a set difference. In classical logic this would be equality rather than subset. (Contributed by Jim Kingdon, 27-Jul-2018.)

Assertion

Ref	Expression
difdif2ss	|- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B \ C ) )

Proof of Theorem difdif2ss

Step	Hyp	Ref	Expression
1		inssdif 3443	. . . 4 |- ( A i^i C ) C_ ( A \ ( _V \ C ) )
2		unss2 3378	. . . 4 |- ( ( A i^i C ) C_ ( A \ ( _V \ C ) ) -> ( ( A \ B ) u. ( A i^i C ) ) C_ ( ( A \ B ) u. ( A \ ( _V \ C ) ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( ( A \ B ) u. ( A \ ( _V \ C ) ) )
4		difindiss 3461	. . 3 |- ( ( A \ B ) u. ( A \ ( _V \ C ) ) ) C_ ( A \ ( B i^i ( _V \ C ) ) )
5	3, 4	sstri 3236	. 2 |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B i^i ( _V \ C ) ) )
6		invdif 3449	. . . 4 |- ( B i^i ( _V \ C ) ) = ( B \ C )
7	6	eqcomi 2235	. . 3 |- ( B \ C ) = ( B i^i ( _V \ C ) )
8	7	difeq2i 3322	. 2 |- ( A \ ( B \ C ) ) = ( A \ ( B i^i ( _V \ C ) ) )
9	5, 8	sseqtrri 3262	1 |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B \ C ) )

Syntax hints:   _Vcvv 2802   \ cdif 3197   u. cun 3198   i^i cin 3199   C_ wss 3200

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213

This theorem is referenced by: (None)