Theorem ch0le 30091

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 29874	. 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		sh0le 30090	. 2 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3902   S_ℋ csh 29578   C_ℋ cch 29579  0_ℋc0h 29585

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5248  ax-hilex 29649

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-br 5098  df-opab 5160  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6436  df-fv 6492  df-ov 7345  df-sh 29857  df-ch 29871  df-ch0 29903

This theorem is referenced by:  chnlen0  30094  ch0pss  30095  ch0lei  30101  chssoc  30146  atcveq0  30998