Theorem cldss 21634

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 21631	. 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		iscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld 21632	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
4	3	simprbda 502	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ⊆ 𝑋)
5	1, 4	mpancom 687	1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ∖ cdif 3878   ⊆ wss 3881  ∪ cuni 4800  ‘cfv 6324  Topctop 21498  Clsdccld 21621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-top 21499  df-cld 21624

This theorem is referenced by:  cldss2  21635  iincld  21644  uncld  21646  cldcls  21647  iuncld  21650  clsval2  21655  clsss3  21664  clsss2  21677  opncldf1  21689  restcldr  21779  lmcld  21908  nrmsep2  21961  nrmsep  21962  isnrm2  21963  regsep2  21981  cmpcld  22007  dfconn2  22024  conncompclo  22040  cldllycmp  22100  txcld  22208  ptcld  22218  imasncld  22296  kqcldsat  22338  kqnrmlem1  22348  kqnrmlem2  22349  nrmhmph  22399  ufildr  22536  metnrmlem1a  23463  metnrmlem1  23464  metnrmlem2  23465  metnrmlem3  23466  cnheiborlem  23559  cmetss  23920  bcthlem5  23932  cldssbrsiga  31556  clsun  33789  cldregopn  33792  pibt2  34834  mblfinlem3  35096  mblfinlem4  35097  ismblfin  35098  cmpfiiin  39638  kelac1  40007  stoweidlem18  42660  stoweidlem57  42699