Theorem ch0lei 32053

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 32043	. 2 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 0ℋ ⊆ 𝐴

This proof depends on syntax axioms:   ∈ wcel 2145   ⊆ wss 3899   C_ℋ cch 31531  0_ℋc0h 31537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-hilex 31601

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fv 6546  df-ov 7423  df-sh 31809  df-ch 31823  df-ch0 31855

This theorem is used by:  chj0i  32057  chm0i  32092  hst0  32835