# Simple proof

1. Sep 7, 2007

### codi

1. The problem statement, all variables and given/known data

Trying to prove some of the basic laws in set theory, and would like any opinions on 1 of my proofs (eg hints on how can I improve it, is it even a valid proof). Thanks in advance.

$$(A \subseteq B \wedge B \subseteq C) \rightarrow (A \subseteq C)$$

2. Relevant equations

3. The attempt at a solution

1) $$\forall x \in A, x \in B$$ - definition of a subset
2) $$\forall x \in B, x \in C$$ - definition of a subset
3) $$\forall x \in A, x \in C$$ - 1, 2
4) $$A \subseteq C$$ - 3, definition of a subset

2. Sep 7, 2007

### EnumaElish

Suppose A $\nsubseteq$ C. Then there must be some a in A that is not in C. Since B $\subseteq$ C, a cannot be in B. This contradicts A $\subseteq$ B.