Theorem pssnssi

Description: A proper subclass does not include the other class. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Hypothesis	pssnssi.1	⊢ 𝐴 ⊊ 𝐵
Assertion	pssnssi	⊢ ¬ 𝐵 ⊆ 𝐴

Proof

Step	Hyp	Ref	Expression
1		pssnssi.1	⊢ 𝐴 ⊊ 𝐵
2		dfpss3	⊢ ( 𝐴 ⊊ 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴 ) )
3	1 2	mpbi	⊢ ( 𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴 )
4	3	simpri	⊢ ¬ 𝐵 ⊆ 𝐴