Theorem chsssh 30983

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 30982	. 2 ⊢ (𝑥 ∈ Cℋ → 𝑥 ∈ Sℋ )
2	1	ssriv 3981	1 ⊢ Cℋ ⊆ Sℋ

Syntax hints:   ⊆ wss 3943   S_ℋ csh 30686   C_ℋ cch 30687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fv 6544  df-ov 7407  df-ch 30979

This theorem is referenced by:  chex  30984  chsspwh  31005  chintcli  31089  shatomistici  32119