Theorem ssex

Description: A subclass of a set is a set. Exercise 3 of TakeutiZaring p. 22. This is one way to express the Axiom of Separation ax-sep (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994) (Proof shortened by BJ, 18-Jul-2026)

	Ref	Expression
Hypothesis	ssex.1	$$⊢B\in \mathrm{V}$$
Assertion	ssex	$$⊢A\subseteq B\to A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		ssex.1	$$⊢B\in \mathrm{V}$$
2		ssexg	$$⊢\left(A\subseteq B\wedge B\in \mathrm{V}\right)\to A\in \mathrm{V}$$
3	1 2	mpan2	$$⊢A\subseteq B\to A\in \mathrm{V}$$