Theorem ch0lei 29532

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 29522	. 2 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 0ℋ ⊆ 𝐴

Syntax hints:   ∈ wcel 2110   ⊆ wss 3866   C_ℋ cch 29010  0_ℋc0h 29016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-hilex 29080

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fv 6388  df-ov 7216  df-sh 29288  df-ch 29302  df-ch0 29334

This theorem is referenced by:  chj0i  29536  chm0i  29571  hst0  30314