**Proof**: *(related to "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.

q.e.d

| | | | created: 2019-01-13 01:28:34 | modified: 2019-01-13 01:29:19 | by: *bookofproofs* | references: [656], [983]

(none)

[983] **Ebbinghaus, H.-D.**: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3

[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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