Theorem inundifss 3525

Description: The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

Ref	Expression
inundifss	|- ( ( A i^i B ) u. ( A \ B ) ) C_ A

Proof of Theorem inundifss

Step	Hyp	Ref	Expression
1		inss1 3380	. 2 |- ( A i^i B ) C_ A
2		difss 3286	. 2 |- ( A \ B ) C_ A
3	1, 2	unssi 3335	1 |- ( ( A i^i B ) u. ( A \ B ) ) C_ A

Syntax hints:   \ cdif 3151   u. cun 3152   i^i cin 3153   C_ wss 3154

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167

This theorem is referenced by:  resasplitss  5434