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Mirrors > Home > MPE Home > Th. List > difopn | Structured version Visualization version GIF version |
Description: The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.) |
Ref | Expression |
---|---|
iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
difopn | ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 4934 | . . . . . 6 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
2 | iscld.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | sseqtrrdi 4028 | . . . . 5 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋) |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ⊆ 𝑋) |
5 | df-ss 3960 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∩ 𝑋) = 𝐴) | |
6 | 4, 5 | sylib 217 | . . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝑋) = 𝐴) |
7 | 6 | difeq1d 4116 | . 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) = (𝐴 ∖ 𝐵)) |
8 | indif2 4265 | . . 3 ⊢ (𝐴 ∩ (𝑋 ∖ 𝐵)) = ((𝐴 ∩ 𝑋) ∖ 𝐵) | |
9 | cldrcl 22881 | . . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
10 | 9 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top) |
11 | simpl 482 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ 𝐽) | |
12 | 2 | cldopn 22886 | . . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽) |
13 | 12 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝐵) ∈ 𝐽) |
14 | inopn 22752 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽) | |
15 | 10, 11, 13, 14 | syl3anc 1368 | . . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽) |
16 | 8, 15 | eqeltrrid 2832 | . 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) ∈ 𝐽) |
17 | 7, 16 | eqeltrrd 2828 | 1 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∖ cdif 3940 ∩ cin 3942 ⊆ wss 3943 ∪ cuni 4902 ‘cfv 6536 Topctop 22746 Clsdccld 22871 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fn 6539 df-fv 6544 df-top 22747 df-cld 22874 |
This theorem is referenced by: bcthlem5 25207 cldssbrsiga 33715 pibt2 36805 poimirlem30 37029 dirkercncflem2 45373 fourierdlem62 45437 |
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