Theorem isopn2 23170

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 4091	. . . 4 ⊢ (𝑋 ∖ 𝑆) ⊆ 𝑋
2		iscld.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
3	2	iscld2 23166	. . . 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 4223	. . . . 5 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
6	5	biimpi 219	. . . 4 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ (𝑋 ∖ 𝑆)) = 𝑆)
7	6	eleq1d 2848	. . 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 1570   ∈ wcel 2143   ∖ cdif 3903   ⊆ wss 3906  ∪ cuni 4873  ‘cfv 6538  Topctop 23031  Clsdccld 23154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pow 5338  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-top 23032  df-cld 23157

This theorem is referenced by:  opncld  23171  iscncl  23407  1stckgen  23692  txkgen  23790  qtoprest  23855  qtopcmap  23857  stoweidlem28  46725