Theorem sh0le 28992

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 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)

Syntax hints:   → wi 4   ∈ wcel 2048   ⊆ wss 3828  {csn 4439  0_ℎc0v 28474   S_ℋ csh 28478  0_ℋc0h 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