Theorem sh0le 31441

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 31254	. 2 ⊢ 0ℋ = {0ℎ}
2		sh0 31217	. . 3 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴)
3	2	snssd 4762	. 2 ⊢ (𝐴 ∈ Sℋ → {0ℎ} ⊆ 𝐴)
4	1, 3	eqsstrid 3969	1 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2113   ⊆ wss 3898  {csn 4577  0_ℎc0v 30925   S_ℋ csh 30929  0_ℋc0h 30936

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-hilex 31000

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-xp 5627  df-cnv 5629  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-sh 31208  df-ch0 31254

This theorem is referenced by:  ch0le  31442  shle0  31443  orthin  31447  ssjo  31448  shs0i  31450  span0  31543