Theorem sh0le 31526

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 31339	. 2 ⊢ 0ℋ = {0ℎ}
2		sh0 31302	. . 3 ⊢ (𝐴 ∈ Sℋ → 0ℎ ∈ 𝐴)
3	2	snssd 4753	. 2 ⊢ (𝐴 ∈ Sℋ → {0ℎ} ⊆ 𝐴)
4	1, 3	eqsstrid 3961	1 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2114   ⊆ wss 3890  {csn 4568  0_ℎc0v 31010   S_ℋ csh 31014  0_ℋc0h 31021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-hilex 31085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-sh 31293  df-ch0 31339

This theorem is referenced by:  ch0le  31527  shle0  31528  orthin  31532  ssjo  31533  shs0i  31535  span0  31628