Theorem isopn2 23040

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 4136	. . . 4 ⊢ (𝑋 ∖ 𝑆) ⊆ 𝑋
2		iscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld2 23036	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽))
4	1, 3	mpan2 691	. . 3 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽))
5		dfss4 4269	. . . . 5 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
6	5	biimpi 216	. . . 4 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
7	6	eleq1d 2826	. . 3 ⊢ (𝑆 ⊆ 𝑋 → ((𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽 ↔ 𝑆 ∈ 𝐽))
8	4, 7	sylan9bb 509	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ 𝑆 ∈ 𝐽))
9	8	bicomd 223	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∖ cdif 3948   ⊆ wss 3951  ∪ cuni 4907  ‘cfv 6561  Topctop 22899  Clsdccld 23024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-top 22900  df-cld 23027

This theorem is referenced by:  opncld  23041  iscncl  23277  1stckgen  23562  txkgen  23660  qtoprest  23725  qtopcmap  23727  stoweidlem28  46043