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Mirrors > Home > MPE Home > Th. List > Mathboxes > nsstr | Structured version Visualization version GIF version |
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 |
---|---|
nsstr | ⊢ ((¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstr 3974 | . . . 4 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐴 ⊆ 𝐵) | |
2 | 1 | ancoms 461 | . . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ 𝐵) |
3 | 2 | adantll 712 | . 2 ⊢ (((¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ 𝐵) |
4 | simpll 765 | . 2 ⊢ (((¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) ∧ 𝐴 ⊆ 𝐶) → ¬ 𝐴 ⊆ 𝐵) | |
5 | 3, 4 | pm2.65da 815 | 1 ⊢ ((¬ 𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) → ¬ 𝐴 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ⊆ wss 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3942 df-ss 3951 |
This theorem is referenced by: mbfpsssmf 43053 |
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