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.
[tex](A \subseteq B \wedge B \subseteq C) \rightarrow (A \subseteq C)[/tex]
The Attempt at a Solution
1) [tex]\forall x \in A, x \in B$[/tex] - definition of a subset
2) [tex]\forall x \in B, x \in C[/tex] - definition of a subset
3) [tex]\forall x \in A, x \in C[/tex] - 1, 2
4) [tex]A \subseteq C[/tex] - 3, definition of a subset