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Mirrors > Home > MPE Home > Th. List > uncld | Structured version Visualization version GIF version |
Description: The union of two closed sets is closed. Equivalent to Theorem 6.1(3) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) |
Ref | Expression |
---|---|
uncld | ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difundi 4278 | . . 3 ⊢ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) = ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) | |
2 | cldrcl 22991 | . . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
3 | eqid 2725 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | 3 | cldopn 22996 | . . . 4 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ 𝐽) |
5 | 3 | cldopn 22996 | . . . 4 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) |
6 | inopn 22862 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝐴) ∈ 𝐽 ∧ (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽) | |
7 | 2, 4, 5, 6 | syl2an3an 1419 | . . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽) |
8 | 1, 7 | eqeltrid 2829 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽) |
9 | 3 | cldss 22994 | . . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽) |
10 | 3 | cldss 22994 | . . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽) |
11 | 9, 10 | anim12i 611 | . . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽)) |
12 | unss 4182 | . . . 4 ⊢ ((𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽) ↔ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽) | |
13 | 11, 12 | sylib 217 | . . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽) |
14 | 3 | iscld2 22993 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽)) |
15 | 2, 13, 14 | syl2an2r 683 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽)) |
16 | 8, 15 | mpbird 256 | 1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 ∪ cuni 4909 ‘cfv 6549 Topctop 22856 Clsdccld 22981 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fn 6552 df-fv 6557 df-top 22857 df-cld 22984 |
This theorem is referenced by: iscldtop 23060 paste 23259 lpcls 23329 dvasin 37328 dvacos 37329 dvreasin 37330 dvreacos 37331 |
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