The Empty Set: A subset we can all contain

I was just reading Introductory Real Analysis, and the first chapter is on set theory. When introducing the empty set, it states,

“The [empty set] is clearly a subset of every set (why?).”

I have known this fact for a while now, but I hadn't ever been prompted to justify it. It also reminded me for a moment why I generally hate it when math books which I have purchased for reference leaves the justification as an exercise for the reader. As soon as I had finished that thought, the solution came to me. I wanted to express it as simply and rigorously as possible, in hopes that it may clarify this core fact of set theory.

When justifying any claim in mathematics, especially if I get stuck, I try to recall the necessary definitions. In this case, a subset is defined as such:
 \[ A \subseteq B \Leftrightarrow \forall x\left(x \in A \Rightarrow x \in B \right)\]

and the empty set:

\[ \emptyset \equiv \left\{ x | x \notin \emptyset \right\} \]

Now, let's recall the truth table for implication. Every combination of \(X \Rightarrow Y\) evaluates to true except when the antecedent is true and the consequent is false:

• \(\bot \Rightarrow \mathcal{X}  \lor \top \Rightarrow \top \equiv \top\)
• \(\top \Rightarrow \bot \equiv \bot\)

To show that the empty set is a subset of all sets, we only need to look to the lines 1 and 2 of the truth table. Since \(\forall x \left( x \notin \emptyset \right)\) by definition, \(x \in \emptyset\) is always a false, we can substitue \(\emptyset\) into the subset definition:
\[ \emptyset \subseteq B \Leftrightarrow \forall x\left(x \in \emptyset \Rightarrow x \in B\right)\]
which means
\[ \emptyset \subseteq B \Leftrightarrow \forall x\left(\bot \Rightarrow x \in B\right).\]
So, it doesn't matter if \(x \in B\) or \(x \notin B\), because \(\bot \Rightarrow x \in B\) is true regardless of the truth value of \(x \in B\) . Therefore, the emtpy set is a subset of all sets. QED