Theorem psstr

Description: Transitive law for proper subclass. Theorem 9 of Suppes p. 23. (Contributed by NM, 7-Feb-1996)

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

Proof

Step	Hyp	Ref	Expression
1		pssss	⊢ ( 𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵 )
2		pssss	⊢ ( 𝐵 ⊊ 𝐶 → 𝐵 ⊆ 𝐶 )
3	1 2	sylan9ss	⊢ ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → 𝐴 ⊆ 𝐶 )
4		pssn2lp	⊢ ¬ ( 𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 )
5		psseq1	⊢ ( 𝐴 = 𝐶 → ( 𝐴 ⊊ 𝐵 ↔ 𝐶 ⊊ 𝐵 ) )
6	5	anbi1d	⊢ ( 𝐴 = 𝐶 → ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) ↔ ( 𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) ) )
7	4 6	mtbiri	⊢ ( 𝐴 = 𝐶 → ¬ ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) )
8	7	con2i	⊢ ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → ¬ 𝐴 = 𝐶 )
9		dfpss2	⊢ ( 𝐴 ⊊ 𝐶 ↔ ( 𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶 ) )
10	3 8 9	sylanbrc	⊢ ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → 𝐴 ⊊ 𝐶 )