MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isopn2 Structured version   Visualization version   GIF version

Theorem isopn2 21639
Description: A subset of the underlying set of a topology is open iff its complement is closed. (Contributed by NM, 4-Oct-2006.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
isopn2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))

Proof of Theorem isopn2
StepHypRef Expression
1 difss 4107 . . . 4 (𝑋𝑆) ⊆ 𝑋
2 iscld.1 . . . . 5 𝑋 = 𝐽
32iscld2 21635 . . . 4 ((𝐽 ∈ Top ∧ (𝑋𝑆) ⊆ 𝑋) → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋𝑆)) ∈ 𝐽))
41, 3mpan2 689 . . 3 (𝐽 ∈ Top → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋𝑆)) ∈ 𝐽))
5 dfss4 4234 . . . . 5 (𝑆𝑋 ↔ (𝑋 ∖ (𝑋𝑆)) = 𝑆)
65biimpi 218 . . . 4 (𝑆𝑋 → (𝑋 ∖ (𝑋𝑆)) = 𝑆)
76eleq1d 2897 . . 3 (𝑆𝑋 → ((𝑋 ∖ (𝑋𝑆)) ∈ 𝐽𝑆𝐽))
84, 7sylan9bb 512 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ 𝑆𝐽))
98bicomd 225 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  cdif 3932  wss 3935   cuni 4837  cfv 6354  Topctop 21500  Clsdccld 21623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-top 21501  df-cld 21626
This theorem is referenced by:  opncld  21640  iscncl  21876  1stckgen  22161  txkgen  22259  qtoprest  22324  qtopcmap  22326  stoweidlem28  42312
  Copyright terms: Public domain W3C validator