Theorem sspsstrd

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr . (Contributed by David Moews, 1-May-2017)

	Ref	Expression
Hypotheses	sspsstrd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	sspsstrd.2	⊢ ( 𝜑 → 𝐵 ⊊ 𝐶 )
Assertion	sspsstrd	⊢ ( 𝜑 → 𝐴 ⊊ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sspsstrd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		sspsstrd.2	⊢ ( 𝜑 → 𝐵 ⊊ 𝐶 )
3		sspsstr	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → 𝐴 ⊊ 𝐶 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 𝐴 ⊊ 𝐶 )