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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 28803 | . 2 ⊢ 0ℋ = {0ℎ} | |
2 | sh0 28766 | . . 3 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴) | |
3 | 2 | snssd 4614 | . 2 ⊢ (𝐴 ∈ Sℋ → {0ℎ} ⊆ 𝐴) |
4 | 1, 3 | syl5eqss 3904 | 1 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2048 ⊆ wss 3828 {csn 4439 0ℎc0v 28474 Sℋ csh 28478 0ℋc0h 28485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2747 ax-sep 5058 ax-hilex 28549 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-rab 3094 df-v 3414 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-op 4446 df-br 4928 df-opab 4990 df-xp 5410 df-cnv 5412 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-sh 28757 df-ch0 28803 |
This theorem is referenced by: ch0le 28993 shle0 28994 orthin 28998 ssjo 28999 shs0i 29001 span0 29094 |
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