Proving Set Theory Basics: A \subseteq C

[codi]

Homework Statement

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.

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

Homework Equations

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

 

[EnumaElish]

Your logic is correct.

You could also use proof by contradiction:

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.