Theorem sh0le 31369

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 31182	. 2 ⊢ 0ℋ = {0ℎ}
2		sh0 31145	. . 3 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴)
3	2	snssd 4773	. 2 ⊢ (𝐴 ∈ Sℋ → {0ℎ} ⊆ 𝐴)
4	1, 3	eqsstrid 3985	1 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109   ⊆ wss 3914  {csn 4589  0_ℎc0v 30853   S_ℋ csh 30857  0_ℋc0h 30864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-hilex 30928

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-sh 31136  df-ch0 31182

This theorem is referenced by:  ch0le  31370  shle0  31371  orthin  31375  ssjo  31376  shs0i  31378  span0  31471