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Mirrors > Home > ILE Home > Th. List > difopn | 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 3863 | . . . . . 6 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
2 | iscld.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | sseqtrrdi 3228 | . . . . 5 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋) |
4 | 3 | adantr 276 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ⊆ 𝑋) |
5 | df-ss 3166 | . . . 4 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∩ 𝑋) = 𝐴) | |
6 | 4, 5 | sylib 122 | . . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝑋) = 𝐴) |
7 | 6 | difeq1d 3276 | . 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) = (𝐴 ∖ 𝐵)) |
8 | indif2 3403 | . . 3 ⊢ (𝐴 ∩ (𝑋 ∖ 𝐵)) = ((𝐴 ∩ 𝑋) ∖ 𝐵) | |
9 | cldrcl 14270 | . . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
10 | 9 | adantl 277 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top) |
11 | simpl 109 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ 𝐽) | |
12 | 2 | cldopn 14275 | . . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽) |
13 | 12 | adantl 277 | . . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝐵) ∈ 𝐽) |
14 | inopn 14171 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽) | |
15 | 10, 11, 13, 14 | syl3anc 1249 | . . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽) |
16 | 8, 15 | eqeltrrid 2281 | . 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) ∈ 𝐽) |
17 | 7, 16 | eqeltrrd 2271 | 1 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ∖ cdif 3150 ∩ cin 3152 ⊆ wss 3153 ∪ cuni 3835 ‘cfv 5254 Topctop 14165 Clsdccld 14260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fn 5257 df-fv 5262 df-top 14166 df-cld 14263 |
This theorem is referenced by: (None) |
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