Theorem inssddif 3422

Description: Intersection of two classes and class difference. In classical logic, such as Exercise 4.10(q) of [Mendelson] p. 231, this is an equality rather than subset. (Contributed by Jim Kingdon, 26-Jul-2018.)

Assertion

Ref	Expression
inssddif	|- ( A i^i B ) C_ ( A \ ( A \ B ) )

Proof of Theorem inssddif

Step	Hyp	Ref	Expression
1		inss1 3401	. . 3 |- ( A i^i B ) C_ A
2		ssddif 3415	. . 3 |- ( ( A i^i B ) C_ A <-> ( A i^i B ) C_ ( A \ ( A \ ( A i^i B ) ) ) )
3	1, 2	mpbi 145	. 2 |- ( A i^i B ) C_ ( A \ ( A \ ( A i^i B ) ) )
4		difin 3418	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
5	4	difeq2i 3296	. 2 |- ( A \ ( A \ ( A i^i B ) ) ) = ( A \ ( A \ B ) )
6	3, 5	sseqtri 3235	1 |- ( A i^i B ) C_ ( A \ ( A \ B ) )

Syntax hints:   \ cdif 3171   i^i cin 3173   C_ wss 3174

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rab 2495  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187

This theorem is referenced by: (None)