Theorem inssddif 3363

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 3342	. . 3 |- ( A i^i B ) C_ A
2		ssddif 3356	. . 3 |- ( ( A i^i B ) C_ A <-> ( A i^i B ) C_ ( A \ ( A \ ( A i^i B ) ) ) )
3	1, 2	mpbi 144	. 2 |- ( A i^i B ) C_ ( A \ ( A \ ( A i^i B ) ) )
4		difin 3359	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
5	4	difeq2i 3237	. 2 |- ( A \ ( A \ ( A i^i B ) ) ) = ( A \ ( A \ B ) )
6	3, 5	sseqtri 3176	1 |- ( A i^i B ) C_ ( A \ ( A \ B ) )

Syntax hints:   \ cdif 3113   i^i cin 3115   C_ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129

This theorem is referenced by: (None)