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 29219 | . . 3 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) | |
2 | 1 | biantrud 534 | . 2 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴))) |
3 | eqss 3984 | . 2 ⊢ (𝐴 = 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴)) | |
4 | 2, 3 | syl6bbr 291 | 1 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 Sℋ csh 28707 0ℋc0h 28714 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-hilex 28778 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-sh 28986 df-ch0 29032 |
This theorem is referenced by: chle0 29222 shne0i 29227 shs00i 29229 cdj3lem1 30213 |
Copyright terms: Public domain | W3C validator |