Theorem cldss 21640

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 21637	. 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		iscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld 21638	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
4	3	simprbda 501	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ⊆ 𝑋)
5	1, 4	mpancom 686	1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113   ∖ cdif 3936   ⊆ wss 3939  ∪ cuni 4841  ‘cfv 6358  Topctop 21504  Clsdccld 21627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366  df-top 21505  df-cld 21630

This theorem is referenced by:  cldss2  21641  iincld  21650  uncld  21652  cldcls  21653  iuncld  21656  clsval2  21661  clsss3  21670  clsss2  21683  opncldf1  21695  restcldr  21785  lmcld  21914  nrmsep2  21967  nrmsep  21968  isnrm2  21969  regsep2  21987  cmpcld  22013  dfconn2  22030  conncompclo  22046  cldllycmp  22106  txcld  22214  ptcld  22224  imasncld  22302  kqcldsat  22344  kqnrmlem1  22354  kqnrmlem2  22355  nrmhmph  22405  ufildr  22542  metnrmlem1a  23469  metnrmlem1  23470  metnrmlem2  23471  metnrmlem3  23472  cnheiborlem  23561  cmetss  23922  bcthlem5  23934  cldssbrsiga  31450  clsun  33680  cldregopn  33683  pibt2  34702  mblfinlem3  34935  mblfinlem4  34936  ismblfin  34937  cmpfiiin  39300  kelac1  39669  stoweidlem18  42310  stoweidlem57  42349