# Set Theory Question

Tags:
1. Oct 11, 2016

1. The problem statement, all variables and given/known data
$C \subseteq A \cap B \implies A \cap B \cap C = C$

2. Relevant equations
How do I get rid of the "belongs to" term on the right hand side? I know I need to prove either the left hand or the right hand side of the "or" term is correct, I'm just not sure how to get there.

3. The attempt at a solution
~$(C \subseteq A \cap B) \cup (A \cap B \cap C = C)$

right hand side (right of the "or"):
$C \subseteq A \cap B \cap C$ (Trivial)
$A \cap B \cap C \subseteq C$ (This is the one we want to prove)

So all together:

~$(C \subseteq A \cap B) \cup (A \cap B \cap C \subseteq C)$
$\exists x \in C \therefore x \in A \cap B)$
$(\sim a \cup \sim b) \cup (a \cap b \cap c \subseteq C)$

2. Oct 11, 2016

### Staff: Mentor

I don't really understand your complexity here. Can't you simply use $(X \subseteq Y) \wedge (Y \subseteq X) \Longrightarrow X = Y$?