Theorem psssstrd

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

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

Proof

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