Theorem inssddif 3312

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 3291	. . 3 |- ( A i^i B ) C_ A
2		ssddif 3305	. . 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 3308	. . 3 |- ( A \ ( A i^i B ) ) = ( A \ B )
5	4	difeq2i 3186	. 2 |- ( A \ ( A \ ( A i^i B ) ) ) = ( A \ ( A \ B ) )
6	3, 5	sseqtri 3126	1 |- ( A i^i B ) C_ ( A \ ( A \ B ) )

Syntax hints:   \ cdif 3063   i^i cin 3065   C_ wss 3066

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079

This theorem is referenced by: (None)