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

Theorem isopn2 23027
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 4131 . . . 4 (𝑋𝑆) ⊆ 𝑋
2 iscld.1 . . . . 5 𝑋 = 𝐽
32iscld2 23023 . . . 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 4260 . . . . 5 (𝑆𝑋 ↔ (𝑋 ∖ (𝑋𝑆)) = 𝑆)
65biimpi 215 . . . 4 (𝑆𝑋 → (𝑋 ∖ (𝑋𝑆)) = 𝑆)
76eleq1d 2811 . . 3 (𝑆𝑋 → ((𝑋 ∖ (𝑋𝑆)) ∈ 𝐽𝑆𝐽))
84, 7sylan9bb 508 . 2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝑋𝑆) ∈ (Clsd‘𝐽) ↔ 𝑆𝐽))
98bicomd 222 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆𝐽 ↔ (𝑋𝑆) ∈ (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  cdif 3944  wss 3947   cuni 4913  cfv 6554  Topctop 22886  Clsdccld 23011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-top 22887  df-cld 23014
This theorem is referenced by:  opncld  23028  iscncl  23264  1stckgen  23549  txkgen  23647  qtoprest  23712  qtopcmap  23714  stoweidlem28  45649
  Copyright terms: Public domain W3C validator