Theorem ch0le 5373

Description: The closed subspace zero is the smallest member of CH.

Hypothesis

Ref	Expression
ch0le.1	|- A e. CH

Assertion

Ref	Expression
ch0le	|- 0H (_ A

Proof of Theorem ch0le

Step	Hyp	Ref	Expression
1		ch0le.1	. 2 |- A e. CH
2		ch0let 5366	. 2 |- (A e. CH -> 0H (_ A)
3	1, 2	ax-mp 6	1 |- 0H (_ A

Syntax hints:   e. wcel 1092   (_ wss 1487  CHcch 4968  0Hc0h 4974

This theorem is referenced by:  chj0 5377  chm0 5411

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sn 1811  df-sh 5114  df-ch 5127  df-ch0 5157

metamath.org