Mendelson's 4.5 Proposition There is a proposition:
$M(X) \Rightarrow \exists(Z), M(Z) \land X\in Z$
I know how to prove it using Pairing Axiom. ($X\in\{X,X\}$ and $M(\{X,X\})$)
But I saw somewhere the different proof, without Pairing Axiom. 
What can it be? 

$M(X):=$"X is a set"$=\exists Y,X\in Y$.
Both variants are good and worth the upvotes, but I saw a different proof. (It was really great and, probably, no axioms from set theory was used. Just predicate calculus axioms, Bernays rules, Modus Ponens and magic. )


If you really want to avoid the Pairing axiom, you can use Replacement and Infinity to form
$$ \{ F(y) \mid y\in\omega \} $$
where $F(y)=X$ for every $y$.

Alternatively, as martin.koeberl points, out $\mathcal P(X)$ is always a set that has $X$ as an element.
