Theorem ssex

Description: The subset of a set is also 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)

	Ref	Expression
Hypothesis	ssex.1	⊢ 𝐵 ∈ V
Assertion	ssex	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ∈ V )

Proof

Step	Hyp	Ref	Expression
1		ssex.1	⊢ 𝐵 ∈ V
2		df-ss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∩ 𝐵 ) = 𝐴 )
3	1	inex2	⊢ ( 𝐴 ∩ 𝐵 ) ∈ V
4		eleq1	⊢ ( ( 𝐴 ∩ 𝐵 ) = 𝐴 → ( ( 𝐴 ∩ 𝐵 ) ∈ V ↔ 𝐴 ∈ V ) )
5	3 4	mpbii	⊢ ( ( 𝐴 ∩ 𝐵 ) = 𝐴 → 𝐴 ∈ V )
6	2 5	sylbi	⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ∈ V )