![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > shle0 | Structured version Visualization version GIF version |
Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shle0 | ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sh0le 31126 | . . 3 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) | |
2 | 1 | biantrud 531 | . 2 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴))) |
3 | eqss 3997 | . 2 ⊢ (𝐴 = 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴)) | |
4 | 2, 3 | bitr4di 289 | 1 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 Sℋ csh 30614 0ℋc0h 30621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-sep 5299 ax-hilex 30685 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-sh 30893 df-ch0 30939 |
This theorem is referenced by: chle0 31129 shne0i 31134 shs00i 31136 cdj3lem1 32120 |
Copyright terms: Public domain | W3C validator |