Theorem isopn2 23154

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 ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)))

Proof of Theorem isopn2

Step	Hyp	Ref	Expression
1		difss 4098	. . . 4 ⊢ (𝑋 ∖ 𝑆) ⊆ 𝑋
2		iscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld2 23150	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽))
4	1, 3	mpan2 703	. . 3 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽))
5		dfss4 4230	. . . . 5 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
6	5	biimpi 219	. . . 4 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
7	6	eleq1d 2854	. . 3 ⊢ (𝑆 ⊆ 𝑋 → ((𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽 ↔ 𝑆 ∈ 𝐽))
8	4, 7	sylan9bb 518	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ 𝐽))
9	8	bicomd 226	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∖ cdif 3910   ⊆ wss 3913  ∪ cuni 4873  ‘cfv 6533  Topctop 23015  Clsdccld 23138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pow 5334  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6535  df-fv 6541  df-top 23016  df-cld 23141

This theorem is referenced by:  opncld  23155  iscncl  23391  1stckgen  23676  txkgen  23774  qtoprest  23839  qtopcmap  23841  stoweidlem28  46627