Theorem nsstr

Description: If it's not a subclass, it's not a subclass of a smaller one. (Contributed by Glauco Siliprandi, 26-Jun-2021)

	Ref	Expression
Assertion	nsstr	⊢ ( ( ¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → ¬ 𝐴 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sstr	⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵 ) → 𝐴 ⊆ 𝐵 )
2	1	ancoms	⊢ ( ( 𝐶 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶 ) → 𝐴 ⊆ 𝐵 )
3	2	adantll	⊢ ( ( ( ¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵 ) ∧ 𝐴 ⊆ 𝐶 ) → 𝐴 ⊆ 𝐵 )
4		simpll	⊢ ( ( ( ¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵 ) ∧ 𝐴 ⊆ 𝐶 ) → ¬ 𝐴 ⊆ 𝐵 )
5	3 4	pm2.65da	⊢ ( ( ¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵 ) → ¬ 𝐴 ⊆ 𝐶 )