Theorem ch0le 31403

Description: The zero subspace is the smallest member of C_ℋ. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.)

Assertion

Ref	Expression
ch0le	⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)

Proof of Theorem ch0le

Step	Hyp	Ref	Expression
1		chsh 31186	. 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		sh0le 31402	. 2 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2109   ⊆ wss 3905   S_ℋ csh 30890   C_ℋ cch 30891  0_ℋc0h 30897

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 5238  ax-hilex 30961

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 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-xp 5629  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fv 6494  df-ov 7356  df-sh 31169  df-ch 31183  df-ch0 31215

This theorem is referenced by:  chnlen0  31406  ch0pss  31407  ch0lei  31413  chssoc  31458  atcveq0  32310