Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscld2 Structured version   Visualization version   GIF version

Theorem iscld2 20642
 Description: A subset of the underlying set of a topology is closed iff its complement is open. (Contributed by NM, 4-Oct-2006.)
Hypothesis
Ref Expression
iscld.1 𝑋 = 𝐽
Assertion
Ref Expression
iscld2 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑋𝑆) ∈ 𝐽))

Proof of Theorem iscld2
StepHypRef Expression
1 iscld.1 . . 3 𝑋 = 𝐽
21iscld 20641 . 2 (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆𝑋 ∧ (𝑋𝑆) ∈ 𝐽)))
32baibd 946 1 ((𝐽 ∈ Top ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑋𝑆) ∈ 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537   ⊆ wss 3540  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-cld 20633 This theorem is referenced by:  isopn2  20646  0cld  20652  uncld  20655  isclo  20701  cnclima  20882  ist1-2  20961  hausdiag  21258  qtopcld  21326  ufildr  21545  blcld  22120  icccld  22380  iocmnfcld  22382  zcld  22424  recld2  22425  qtophaus  29231  kelac2  36653  stoweidlem50  38943
 Copyright terms: Public domain W3C validator