question about vacuous proof I have a question about vacuous true and it always make me confused. If I want to prove that the empty set is the subset of all the set A, the proof is as following:
if x is in empty set, then x is in A. since x is in empty set is always false,, so the conditional statement is always true~
my question is why x is in empty set is always false, what if x=unicorn, since unicorn is in empty set so x is in empty set is true, right? 
the question goes to why I can not plug a non-exist thing( like unicorn)into the variable x?
 A: You want to prove something like the following:
If $x \in \emptyset$, then $x \in A$.
If you don't like the vacuous nature of the hypothesis, consider the contrapositive (which is logically equivalent to the original statement):
If $x \notin A$, then $x \notin \emptyset$.
Select any $x$ that does not belong to $A$. Surely it also does not belong to the empty set, since the empty set contains no elements at all. (Notice the vacuous hypothesis in the original statement corresponds to a tautological conclusion in the contrapositive.) Having proved the contrapositive, we can also conclude the original statement.
A: @AustinMorh makes the key point. But a footnote about unicorns. We must distinguish two different claims.
Suppose $U$ is the predicate satisfied by all and only unicorns, then

$\forall x(Ux \to x \in \emptyset)$

indeed comes out true for sensible domains (since nothing satisfies the antecedent of the conditional). 'All unicorns are members of the empty set' is another vacuous truth! [But note, we are not "plugging a non-existent thing into a variable": the variables still run over what there is in the domain.]
Suppose however $u$ is now a constant, purportedly naming a unicorn. In standard logic where empty names are not allowed, then $u$ is illegitimate. But take a free logic which does allow empty names. Then atomic wffs involving empty names are not true [on some accounts, they are false, on others they are neither true nor false]. So, since there are no unicorns to be named and $u$ is an empty name, no atomic wff $Fu$ is true, and in particular 

$u \in \emptyset$ 

is not true.
So what are we to make of

if x=unicorn, since unicorn is in empty set so x is in empty set is true?

If x is intended to be name-like, then "x is in empty set" is not true. The truth in the vicinity is something else, i.e. $\forall x(Ux \to x \in \emptyset)$ 
