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 ∈ V
Assertion	ssex	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		ssex.1	⊢ 𝐵 ∈ V
2		ssexg	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V ) → 𝐴 ∈ V )
3	1 2	mpan2	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ∈ V )