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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 30940 | . 2 ⊢ 0ℋ = {0ℎ} | |
2 | sh0 30903 | . . 3 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴) | |
3 | 2 | snssd 4812 | . 2 ⊢ (𝐴 ∈ Sℋ → {0ℎ} ⊆ 𝐴) |
4 | 1, 3 | eqsstrid 4030 | 1 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3948 {csn 4628 0ℎc0v 30611 Sℋ csh 30615 0ℋc0h 30622 |
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 30686 |
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 30894 df-ch0 30940 |
This theorem is referenced by: ch0le 31128 shle0 31129 orthin 31133 ssjo 31134 shs0i 31136 span0 31229 |
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