So I was trying to prove that a null set is a subset of any set.

First, to define when $A$ is a subset of $B$:

$$A \subseteq B \iff \forall x(x \in A \implies x \in B)$$

(At least I think that's right?)

So then consider the empty set $\emptyset$ for which $\forall x(x \not\in \emptyset)$ is true.

I tried to prove that the empty set is a subset of any other set, or:

$$\emptyset \subseteq B \iff \forall x(x \in \emptyset \implies x \in B) \vdash \text{T}$$

To me this seemed true because it was "vacuously true"... somehow. Like it makes sense to call it vacuously true that "all $0$ items in $\emptyset$ can be found in $B$, yep!" but that isn't satisfying to me, how do I "prove" this is the case? 

Is $x \in \emptyset$ a false... statement? A false predicate? Something else? Something that results in false so that the implication itself is true. Is this a $(\text{F}\implies \text{F}) \vdash \text{T}$ thing?

Or is it the $\forall$ that makes it false somehow?

What if I had said $\exists x \in \emptyset$, this feels like it would certainly be false but again I can't prove why, it's just an intuition.

Can anyone clarify the correct definitions / implications and why they're true or false or what have you?