Theorem ch0le 31416

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 31199	. 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ )
2		sh0le 31415	. 2 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴)
3	1, 2	syl 17	1 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3902   S_ℋ csh 30903   C_ℋ cch 30904  0_ℋc0h 30910

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-hilex 30974

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 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fv 6489  df-ov 7349  df-sh 31182  df-ch 31196  df-ch0 31228

This theorem is referenced by:  chnlen0  31419  ch0pss  31420  ch0lei  31426  chssoc  31471  atcveq0  32323