Theorem sh0le 28427

Description: The zero subspace is the smallest subspace. (Contributed by NM, 3-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
sh0le	⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)

Proof of Theorem sh0le

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)

Syntax hints:   → wi 4   ∈ wcel 2030   ⊆ wss 3607  {csn 4210  0_ℎc0v 27909   S_ℋ csh 27913  0_ℋc0h 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