Theorem ch0lei 30564

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 30554	. 2 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 0ℋ ⊆ 𝐴

Syntax hints:   ∈ wcel 2106   ⊆ wss 3943   C_ℋ cch 30042  0_ℋc0h 30048

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 2702  ax-sep 5291  ax-hilex 30112

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5141  df-opab 5203  df-xp 5674  df-cnv 5676  df-dm 5678  df-rn 5679  df-res 5680  df-ima 5681  df-iota 6483  df-fv 6539  df-ov 7395  df-sh 30320  df-ch 30334  df-ch0 30366

This theorem is referenced by:  chj0i  30568  chm0i  30603  hst0  31346