Theorem cldss 22189

Description: A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)

Hypothesis

Ref	Expression
iscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cldss	⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋)

Proof of Theorem cldss

Step	Hyp	Ref	Expression
1		cldrcl 22186	. 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		iscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld 22187	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
4	3	simprbda 499	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ⊆ 𝑋)
5	1, 4	mpancom 685	1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ∖ cdif 3885   ⊆ wss 3888  ∪ cuni 4840  ‘cfv 6437  Topctop 22051  Clsdccld 22176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-iota 6395  df-fun 6439  df-fn 6440  df-fv 6445  df-top 22052  df-cld 22179

This theorem is referenced by:  cldss2  22190  iincld  22199  uncld  22201  cldcls  22202  iuncld  22205  clsval2  22210  clsss3  22219  clsss2  22232  opncldf1  22244  restcldr  22334  lmcld  22463  nrmsep2  22516  nrmsep  22517  isnrm2  22518  regsep2  22536  cmpcld  22562  dfconn2  22579  conncompclo  22595  cldllycmp  22655  txcld  22763  ptcld  22773  imasncld  22851  kqcldsat  22893  kqnrmlem1  22903  kqnrmlem2  22904  nrmhmph  22954  ufildr  23091  metnrmlem1a  24030  metnrmlem1  24031  metnrmlem2  24032  metnrmlem3  24033  cnheiborlem  24126  cmetss  24489  bcthlem5  24501  cldssbrsiga  32164  clsun  34526  cldregopn  34529  pibt2  35597  mblfinlem3  35825  mblfinlem4  35826  ismblfin  35827  cmpfiiin  40526  kelac1  40895  stoweidlem18  43566  stoweidlem57  43605  restcls2lem  46217