TITLE: For every set $A$, the empty set is a subset of $A$. The empty set is a set. Therefore, the empty set has a cardinality $\geq 1\ldots$
QUESTION [6 upvotes]: I have only recently been exposed to sets. According to Wikipedia, as seen on the first bullet mark of the link, $ \forall A: \emptyset \subseteq A$
Does this mean that $\emptyset$ is an element of all sets? (This is False, thank you those that answered)
Is the empty set also a set itself?
Assuming these statements are true, then the empty set therefore an element of the empty set. This does not sound right, please clarify for me. Thank you.

REPLY [2 votes]: Sorry to clutter the site with yet another answer to the many good ones already posted, but I'd like to offer one that is a bit more formal. 
The empty or null set, $\emptyset$, is defined as a set that has no elements. A set $A$ is said to be a subset of $B$ if and only if every element of $A$ is also an element of $B$, notation $A \subseteq B$, or stated using predicate logic,
$$\forall x(x \in A \rightarrow x \in B)$$
So to claim that the null set is a subset of any other set asserts $$\emptyset \in \emptyset \rightarrow \emptyset \in A$$ The logical proof of this simply observes that the antecedent of the implication is false therefore the implication is vacuously true. It is false because the definition of the empty set is one that contains no elements and therefore it does not contain itself. Since we have satisfied the definition of subset, we have proven the claim that the empty set is a subset of every set regardless of how counter intuitive it may be. 
Hey, that's logic!