Theorem chsssh 31205

Description: Closed subspaces are subspaces in a Hilbert space. (Contributed by NM, 29-May-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
chsssh	⊢ Cℋ ⊆ Sℋ

Proof of Theorem chsssh

Step	Hyp	Ref	Expression
1		chsh 31204	. 2 ⊢ (𝑥 ∈ Cℋ → 𝑥 ∈ Sℋ )
2	1	ssriv 3933	1 ⊢ Cℋ ⊆ Sℋ

Syntax hints:   ⊆ wss 3897   S_ℋ csh 30908   C_ℋ cch 30909

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

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-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-ch 31201

This theorem is referenced by:  chex  31206  chsspwh  31227  chintcli  31311  shatomistici  32341