Theorem cldopn 22090

Description: The complement of a closed set is open. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.)

Hypothesis

Ref	Expression
iscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cldopn	⊢ (𝑆 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑆) ∈ 𝐽)

Proof of Theorem cldopn

Step	Hyp	Ref	Expression
1		cldrcl 22085	. 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
2		iscld.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld 22086	. . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽)))
4	3	simplbda 499	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑆) ∈ 𝐽)
5	1, 4	mpancom 684	1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑆) ∈ 𝐽)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880   ⊆ wss 3883  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  Clsdccld 22075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-top 21951  df-cld 22078

This theorem is referenced by:  difopn  22093  iincld  22098  uncld  22100  iuncld  22104  clsval2  22109  opncldf1  22143  opncldf3  22145  restcld  22231  lecldbas  22278  cnclima  22327  nrmsep2  22415  nrmsep  22416  regsep2  22435  cmpcld  22461  dfconn2  22478  txcld  22662  ptcld  22672  kqcldsat  22792  regr1lem  22798  filconn  22942  cldsubg  23170  limcnlp  24947  dvrec  25024  dvexp3  25047  lhop1lem  25082  abelth  25505  logdmopn  25709  lgamucov  26092  onsucconni  34553  onint1  34565  pibt2  35515  mblfinlem3  35743  mblfinlem4  35744  ismblfin  35745  dvtanlem  35753  dvasin  35788  dvacos  35789  dvreasin  35790  dvreacos  35791  fourierdlem62  43599  opncldeqv  46083  iscnrm3rlem5  46126