In this post I will assume the NBG set theory
For all classes $X$ let $M(X)$ mean $(\exists T)(X \in T)$ i.e. $X$ is a set and not a proper class.
Prove that $\left\{X \subseteq Y, M(Y) \right\} \vdash_{\text{NBG}} M(X)$
My ideea for the proof. We know that for all classes $X$ and $Y$ that $X \subseteq Y \implies \mathscr{P}(X) \subseteq \mathscr{P}(Y)$, where $\mathscr{P}(X)$ denotes the power class of $X$ i.e $(\forall z)(z \in \mathscr{P}(X) \iff z \subseteq X)$.
Going back to my original question: $$\left\{X \subseteq Y, M(Y) \right\} \vdash_{\text{NBG}} M(X)$$
$X \subseteq Y \implies X \in \mathscr{P}(Y)$ and $M(Y) \implies M(\mathscr{P}(Y)) \implies M(X)$. The question on the title, namely that the intersection of a set and a class follows immediately, since $A \cap B \subseteq A$
However, in my proof, I used the axiom of the powerset, namely the powerclass of a set is a set. Is there a way to prove my statement(either the one in the title of the one above, intuitively I think they are equivalent but correct me if I am wrong) without using the axiom of the powerset? And if we can't what would be a good proof/justification for why we need to assume such axiom?
P.S.: I am currently studying a book called "Introduction to Mathematical Logic" Sixth Edition by Elliot Mendelson published at CRC Press. Link here page $268$. The book presents the axiomatic system of NBG, however it includes a certain axiom which states that the intersection of a class and a set is a set. It also states that this axiom is a consequence of the standard axioms of NBG, but it can be used to produce weaker versions of NBG.