Theorem ssexOLD

Description: Obsolete version of ssex as of 18-Jul-2026. (Contributed by NM, 27-Apr-1994) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		ssex.1	⊢ 𝐵 ∈ V
2		dfss2	⊢ ( 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 )