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 29144 | . . 3 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) | |
2 | 1 | biantrud 532 | . 2 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴))) |
3 | eqss 3979 | . 2 ⊢ (𝐴 = 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴)) | |
4 | 2, 3 | syl6bbr 290 | 1 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 Sℋ csh 28632 0ℋc0h 28639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-hilex 28703 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-sh 28911 df-ch0 28957 |
This theorem is referenced by: chle0 29147 shne0i 29152 shs00i 29154 cdj3lem1 30138 |
Copyright terms: Public domain | W3C validator |