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Proof

(related to Corollary: Justification of Subsets and Supersets)

• The existence of a superset $X$ is ensured by the axiom of existence.
• The existence of a subset $A\subseteq X$ follows immediately from the axiom of separation. Thus, we have $A=\{z\in X\mid p(z,X_1,\ldots,X_n)\}$ for some logical formula $p(z,X_1,\ldots,X_n).$
• The axiom of extensionality ensures the uniqueness of such a set.

Remark: In particular, a subset can be empty, which is ensured by the axiom of empty set.

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References

Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
2. Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition