Theorem difdif2ss 3488

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 3467	. . . 4 |- ( A i^i C ) C_ ( A \ ( _V \ C ) )
2		unss2 3400	. . . 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 3485	. . 3 |- ( ( A \ B ) u. ( A \ ( _V \ C ) ) ) C_ ( A \ ( B i^i ( _V \ C ) ) )
5	3, 4	sstri 3257	. 2 |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B i^i ( _V \ C ) ) )
6		invdif 3473	. . . 4 |- ( B i^i ( _V \ C ) ) = ( B \ C )
7	6	eqcomi 2242	. . 3 |- ( B \ C ) = ( B i^i ( _V \ C ) )
8	7	difeq2i 3344	. 2 |- ( A \ ( B \ C ) ) = ( A \ ( B i^i ( _V \ C ) ) )
9	5, 8	sseqtrri 3283	1 |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B \ C ) )

Syntax hints:   _Vcvv 2821   \ cdif 3217   u. cun 3218   i^i cin 3219   C_ wss 3220

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233

This theorem is referenced by: (None)