Theorem inindif 4076

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

Assertion

Ref	Expression
inindif	⊢ ((A ∩ B) ∩ (A ∖ B)) = ∅

Proof of Theorem inindif

Step	Hyp	Ref	Expression
1		df-dif 3216	. . 3 ⊢ (A ∖ B) = (A ∩ ∼ B)
2	1	ineq2i 3455	. 2 ⊢ ((A ∩ B) ∩ (A ∖ B)) = ((A ∩ B) ∩ (A ∩ ∼ B))
3		inindi 3473	. 2 ⊢ (A ∩ (B ∩ ∼ B)) = ((A ∩ B) ∩ (A ∩ ∼ B))
4		incompl 4074	. . . 4 ⊢ (B ∩ ∼ B) = ∅
5	4	ineq2i 3455	. . 3 ⊢ (A ∩ (B ∩ ∼ B)) = (A ∩ ∅)
6		in0 3577	. . 3 ⊢ (A ∩ ∅) = ∅
7	5, 6	eqtri 2373	. 2 ⊢ (A ∩ (B ∩ ∼ B)) = ∅
8	2, 3, 7	3eqtr2i 2379	1 ⊢ ((A ∩ B) ∩ (A ∖ B)) = ∅

Syntax hints:   = wceq 1642   ∼ ccompl 3206   ∖ cdif 3207   ∩ cin 3209  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  sbthlem1  6204