Theorem nsstr 45100

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

Assertion

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

Proof of Theorem nsstr

Step	Hyp	Ref	Expression
1		sstr 3992	. . . 4 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
2	1	ancoms 458	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
3	2	adantll 714	. 2 ⊢ (((¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ 𝐵)
4		simpll 767	. 2 ⊢ (((¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) ∧ 𝐴 ⊆ 𝐶) → ¬ 𝐴 ⊆ 𝐵)
5	3, 4	pm2.65da 817	1 ⊢ ((¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ⊆ wss 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-an 396  df-ss 3968

This theorem is referenced by:  mbfpsssmf  46798