Theorem ch0lei 31431

Description: The closed subspace zero is the smallest member of C_ℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ch0le.1	⊢ 𝐴 ∈ Cℋ

Assertion

Ref	Expression
ch0lei	⊢ 0ℋ ⊆ 𝐴

Proof of Theorem ch0lei

Step	Hyp	Ref	Expression
1		ch0le.1	. 2 ⊢ 𝐴 ∈ Cℋ
2		ch0le 31421	. 2 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 0ℋ ⊆ 𝐴

Syntax hints:   ∈ wcel 2111   ⊆ wss 3897   C_ℋ cch 30909  0_ℋc0h 30915

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 5232  ax-hilex 30979

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 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fv 6489  df-ov 7349  df-sh 31187  df-ch 31201  df-ch0 31233

This theorem is referenced by:  chj0i  31435  chm0i  31470  hst0  32213