Theorem isopn2 22856

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 4131	. . . 4 ⊢ (𝑋 ∖ 𝑆) ⊆ 𝑋
2		iscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld2 22852	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽))
4	1, 3	mpan2 688	. . 3 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) ∈ 𝐽))
5		dfss4 4258	. . . . 5 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
6	5	biimpi 215	. . . 4 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
7	6	eleq1d 2817	. . 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 222	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐽 ↔ (𝑋 ∖ 𝑆) ∈ (Clsd‘𝐽)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ∖ cdif 3945   ⊆ wss 3948  ∪ cuni 4908  ‘cfv 6543  Topctop 22715  Clsdccld 22840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-top 22716  df-cld 22843

This theorem is referenced by:  opncld  22857  iscncl  23093  1stckgen  23378  txkgen  23476  qtoprest  23541  qtopcmap  23543  stoweidlem28  45205