Why a set that is subset/equal to infinite set isn't infinite? (by definition) I searched but couldn't really find an answer for that, so sorry if its a duplicate or anything else.
My question is why a set, for example A, that has a subset/equal set which is infinite(N for example), isn't infinite by definition?
The formal definition for an infinite set is "A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number."
It makes sense of course, and its easy to prove that A is also infinite, but then why wouldn't N ⊆ A also mean that A is infinite by definition? Is there an opposite example when it doesn't happen? I'm not very familiar with math definitions in english, so sorry if I wrote something wrong. Thanks.
Edit - @Asaf Karagila answered it perfectly. Thanks for the replies.
 A: Yes. A set with an infinite subset is infinite. But this is not "by definition", but rather a theorem (or a proposition).
In mathematics we have definitions, and we have consequences. The definition of "an infinite set" should not refer to "infinite subsets". In fact, the standard definition of "infinite" is simply "not finite".
But we can prove that $X$ is infinite if and only if it has arbitrarily large finite sets. And then we can easily prove that if $X$ is infinite and $X\subseteq Y$, then $Y$ is infinite.
The thing is that when you say "this holds by definition", then you mean that this is the literal definition (perhaps with a minor and obvious modification). Whereas the definition of an infinite set is not that $\Bbb N$ is a subset of that set, or so on.
A: So your definition of an infinite set would be

A set is infinite if it contains an infinite subset.

This definition cannot be used to prove that any set is infinite. For example, we cannot prove that $\mathbb N$ is infinite, because to prove it is infinite, we need to find an infinite subset of $\mathbb N$. Ok, maybe $A_1=\mathbb N\setminus\{1\}$? 
But how can we prove $A_1$ is infinite? Well, we have to find $A_2$, an infinite subset of $A_1$. And how can we prove $A_2$ is infinite? Well, by finding some subset $A_3$ of $A_2$ which is infinite. OK, but how do we prove $A_3$ is infinite? $\color{red}{\text{and on, and on, and on...}}$
