Theorem sselid

Description: Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014)

	Ref	Expression
Hypotheses	sseli.1	⊢ 𝐴 ⊆ 𝐵
	sselid.2	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
Assertion	sselid	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		sseli.1	⊢ 𝐴 ⊆ 𝐵
2		sselid.2	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
3	1	sseli	⊢ ( 𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵 )
4	2 3	syl	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )