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Mirrors > Home > HSE Home > Th. List > sh0le | Structured version Visualization version GIF version |
Description: The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sh0le | ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ch0 28238 | . 2 ⊢ 0ℋ = {0ℎ} | |
2 | sh0 28201 | . . 3 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴) | |
3 | 2 | snssd 4372 | . 2 ⊢ (𝐴 ∈ Sℋ → {0ℎ} ⊆ 𝐴) |
4 | 1, 3 | syl5eqss 3682 | 1 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 ⊆ wss 3607 {csn 4210 0ℎc0v 27909 Sℋ csh 27913 0ℋc0h 27920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-hilex 27984 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-xp 5149 df-cnv 5151 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-sh 28192 df-ch0 28238 |
This theorem is referenced by: ch0le 28428 shle0 28429 orthin 28433 ssjo 28434 shs0i 28436 span0 28529 |
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