Theorem difdif2ss 3394

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 3373	. . . 4 |- ( A i^i C ) C_ ( A \ ( _V \ C ) )
2		unss2 3308	. . . 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 3391	. . 3 |- ( ( A \ B ) u. ( A \ ( _V \ C ) ) ) C_ ( A \ ( B i^i ( _V \ C ) ) )
5	3, 4	sstri 3166	. 2 |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B i^i ( _V \ C ) ) )
6		invdif 3379	. . . 4 |- ( B i^i ( _V \ C ) ) = ( B \ C )
7	6	eqcomi 2181	. . 3 |- ( B \ C ) = ( B i^i ( _V \ C ) )
8	7	difeq2i 3252	. 2 |- ( A \ ( B \ C ) ) = ( A \ ( B i^i ( _V \ C ) ) )
9	5, 8	sseqtrri 3192	1 |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B \ C ) )

Syntax hints:   _Vcvv 2739   \ cdif 3128   u. cun 3129   i^i cin 3130   C_ wss 3131

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

This theorem is referenced by: (None)