Theorem sspsstr

Description: Transitive law for subclass and proper subclass. (Contributed by NM, 3-Apr-1996)

	Ref	Expression
Assertion	sspsstr	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → 𝐴 ⊊ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sspss	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) )
2		psstr	⊢ ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → 𝐴 ⊊ 𝐶 )
3	2	ex	⊢ ( 𝐴 ⊊ 𝐵 → ( 𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶 ) )
4		psseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶 ) )
5	4	biimprd	⊢ ( 𝐴 = 𝐵 → ( 𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶 ) )
6	3 5	jaoi	⊢ ( ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) → ( 𝐵 ⊊ 𝐶 → 𝐴 ⊊ 𝐶 ) )
7	6	imp	⊢ ( ( ( 𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ) ∧ 𝐵 ⊊ 𝐶 ) → 𝐴 ⊊ 𝐶 )
8	1 7	sylanb	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → 𝐴 ⊊ 𝐶 )