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Mirrors > Home > MPE Home > Th. List > Mathboxes > clddisj | Structured version Visualization version GIF version |
Description: Two ways of saying that two closed sets are disjoint, if 𝐽 is a topology and 𝑋 is a closed set. An alternative proof is similar to that of opndisj 45636 with elssuni 4833 replaced by the combination of cldss 21742 and eqid 2758. (Contributed by Zhi Wang, 6-Sep-2024.) |
Ref | Expression |
---|---|
clddisj | ⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3876 | . 2 ⊢ (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍)) | |
2 | simpl 486 | . . . . 5 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = (∪ 𝐽 ∖ 𝑋)) | |
3 | cldrcl 21739 | . . . . . . 7 ⊢ (𝑌 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
4 | clduni 45634 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽) | |
5 | 4 | difeq1d 4029 | . . . . . . 7 ⊢ (𝐽 ∈ Top → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋)) |
6 | 3, 5 | syl 17 | . . . . . 6 ⊢ (𝑌 ∈ (Clsd‘𝐽) → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋)) |
7 | 6 | adantl 485 | . . . . 5 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (∪ (Clsd‘𝐽) ∖ 𝑋) = (∪ 𝐽 ∖ 𝑋)) |
8 | 2, 7 | eqtr4d 2796 | . . . 4 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋)) |
9 | opndisj 45636 | . . . . . 6 ⊢ (𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅))) | |
10 | 1, 9 | bitr3id 288 | . . . . 5 ⊢ (𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅))) |
11 | 10 | pm5.32dra 45622 | . . . 4 ⊢ ((𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋 ∩ 𝑌) = ∅)) |
12 | 8, 11 | sylancom 591 | . . 3 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝑌 ∈ 𝒫 𝑍 ↔ (𝑋 ∩ 𝑌) = ∅)) |
13 | 12 | pm5.32da 582 | . 2 ⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → ((𝑌 ∈ (Clsd‘𝐽) ∧ 𝑌 ∈ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅))) |
14 | 1, 13 | syl5bb 286 | 1 ⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3857 ∩ cin 3859 ∅c0 4227 𝒫 cpw 4497 ∪ cuni 4801 ‘cfv 6340 Topctop 21606 Clsdccld 21729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-iota 6299 df-fun 6342 df-fn 6343 df-fv 6348 df-mre 16928 df-top 21607 df-topon 21624 df-cld 21732 |
This theorem is referenced by: (None) |
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