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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 45702 with elssuni 4825 replaced by the combination of cldss 21773 and eqid 2738. (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 3857 | . 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 21770 | . . . . . . 7 ⊢ (𝑌 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
4 | clduni 45700 | . . . . . . . 8 ⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽) | |
5 | 4 | difeq1d 4010 | . . . . . . 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 2776 | . . . 4 ⊢ ((𝑍 = (∪ 𝐽 ∖ 𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑍 = (∪ (Clsd‘𝐽) ∖ 𝑋)) |
9 | opndisj 45702 | . . . . . 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 45658 | . . . 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 1542 ∈ wcel 2113 ∖ cdif 3838 ∩ cin 3840 ∅c0 4209 𝒫 cpw 4485 ∪ cuni 4793 ‘cfv 6333 Topctop 21637 Clsdccld 21760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6291 df-fun 6335 df-fn 6336 df-fv 6341 df-mre 16953 df-top 21638 df-topon 21655 df-cld 21763 |
This theorem is referenced by: (None) |
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