Theorem sseldd

Description: Membership inference from subclass relationship. (Contributed by NM, 14-Dec-2004)

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

Proof

Step	Hyp	Ref	Expression
1		sseld.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		sseldd.2	⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
3	1	sseld	⊢ ( 𝜑 → ( 𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵 ) )
4	2 3	mpd	⊢ ( 𝜑 → 𝐶 ∈ 𝐵 )