Expressing an assertion as first-order sentences in ZF Set theory

Question

I want to express the following assertion as first-order sentences in the language of set theory (that is, without abbreviations, in terms of logical symbols and relation symbols $=, \in$ only);

The instance of the Axiom Schema of Separation for the property "$x \subseteq \bigcup$$ x$"

Please can anyone help me out?

Answer 1

Hint

You have to use the definitions:

$A \subseteq B \leftrightarrow \forall z \ (z \in A \to z \in B)$

and;

$z \in \bigcup C \leftrightarrow \exists D \ (z \in D \land D \in C)$.

Thus, $A \subseteq \bigcup A$ will be:

$\forall z \ (z \in A \to \exists D \ (z \in D \land D \in A))$.