Theorem inssddif 3348

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 3327	. . 3 |- ( A i^i B ) C_ A
2		ssddif 3341	. . 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 3344	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
5	4	difeq2i 3222	. 2 |- ( A \ ( A \ ( A i^i B ) ) ) = ( A \ ( A \ B ) )
6	3, 5	sseqtri 3162	1 |- ( A i^i B ) C_ ( A \ ( A \ B ) )

Syntax hints:   \ cdif 3099   i^i cin 3101   C_ wss 3102

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115

This theorem is referenced by: (None)