Theorem inindif 4075

Description: The intersection of an intersection and a difference is empty. (Contributed by set.mm contributors, 10-Mar-2015.)

Assertion

Ref	Expression
inindif	|- ((A i^i B) i^i (A \ B)) = (/)

Proof of Theorem inindif

Step	Hyp	Ref	Expression
1		df-dif 3215	. . 3 |- (A \ B) = (A i^i ∼ B)
2	1	ineq2i 3454	. 2 |- ((A i^i B) i^i (A \ B)) = ((A i^i B) i^i (A i^i ∼ B))
3		inindi 3472	. 2 |- (A i^i (B i^i ∼ B)) = ((A i^i B) i^i (A i^i ∼ B))
4		incompl 4073	. . . 4 |- (B i^i ∼ B) = (/)
5	4	ineq2i 3454	. . 3 |- (A i^i (B i^i ∼ B)) = (A i^i (/))
6		in0 3576	. . 3 |- (A i^i (/)) = (/)
7	5, 6	eqtri 2373	. 2 |- (A i^i (B i^i ∼ B)) = (/)
8	2, 3, 7	3eqtr2i 2379	1 |- ((A i^i B) i^i (A \ B)) = (/)

Syntax hints:   = wceq 1642   ∼ ccompl 3205   \ cdif 3206   i^i cin 3208  (/)c0 3550

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551

This theorem is referenced by:  sbthlem1  6203