Theorem chshii 28424

Description: A closed subspace is a subspace. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

Hypothesis

Ref	Expression
chshi.1	⊢ 𝐻 ∈ Cℋ

Assertion

Ref	Expression
chshii	⊢ 𝐻 ∈ Sℋ

Proof of Theorem chshii

Step	Hyp	Ref	Expression
1		chshi.1	. 2 ⊢ 𝐻 ∈ Cℋ
2		chsh 28421	. 2 ⊢ (𝐻 ∈ Cℋ → 𝐻 ∈ Sℋ )
3	1, 2	ax-mp 5	1 ⊢ 𝐻 ∈ Sℋ

Syntax hints:   ∈ wcel 2145   S_ℋ csh 28125   C_ℋ cch 28126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fv 6038  df-ov 6799  df-ch 28418

This theorem is referenced by:  chssii  28428  helsh  28442  h0elsh  28453  hhsscms  28476  hhssbn  28477  hhsshl  28478  chocunii  28500  shsleji  28569  shjshcli  28575  pjhthlem1  28590  pjhthlem2  28591  omlsii  28602  ococi  28604  pjoc1i  28630  chne0i  28652  chocini  28653  chjcli  28656  chsleji  28657  chseli  28658  chunssji  28666  chjcomi  28667  chub1i  28668  chlubi  28670  chlej1i  28672  chlej2i  28673  h1de2bi  28753  h1de2ctlem  28754  spansnpji  28777  spanunsni  28778  h1datomi  28780  pjoml2i  28784  qlaxr3i  28835  osumi  28841  osumcor2i  28843  spansnji  28845  spansnm0i  28849  nonbooli  28850  spansncvi  28851  5oai  28860  3oalem2  28862  3oalem5  28865  3oalem6  28866  pjaddii  28874  pjmulii  28876  pjss2i  28879  pjssmii  28880  pj0i  28892  pjocini  28897  pjjsi  28899  pjpythi  28921  mayete3i  28927  pjnmopi  29347  pjimai  29375  pjclem4  29398  pj3si  29406  sto1i  29435  stlei  29439  strlem1  29449  hatomici  29558  hatomistici  29561  atomli  29581  chirredlem3  29591  sumdmdii  29614  sumdmdlem  29617  sumdmdlem2  29618