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Mirrors > Home > ILE Home > Th. List > uncld | 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 3328 | . . 3 ⊢ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) = ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) | |
2 | cldrcl 12274 | . . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
3 | 2 | adantr 274 | . . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top) |
4 | eqid 2139 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
5 | 4 | cldopn 12279 | . . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐴) ∈ 𝐽) |
6 | 5 | adantr 274 | . . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝐴) ∈ 𝐽) |
7 | 4 | cldopn 12279 | . . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) |
8 | 7 | adantl 275 | . . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) |
9 | inopn 12173 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (∪ 𝐽 ∖ 𝐴) ∈ 𝐽 ∧ (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽) | |
10 | 3, 6, 8, 9 | syl3anc 1216 | . . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((∪ 𝐽 ∖ 𝐴) ∩ (∪ 𝐽 ∖ 𝐵)) ∈ 𝐽) |
11 | 1, 10 | eqeltrid 2226 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽) |
12 | 4 | cldss 12277 | . . . . 5 ⊢ (𝐴 ∈ (Clsd‘𝐽) → 𝐴 ⊆ ∪ 𝐽) |
13 | 4 | cldss 12277 | . . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽) |
14 | 12, 13 | anim12i 336 | . . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽)) |
15 | unss 3250 | . . . 4 ⊢ ((𝐴 ⊆ ∪ 𝐽 ∧ 𝐵 ⊆ ∪ 𝐽) ↔ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽) | |
16 | 14, 15 | sylib 121 | . . 3 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽) |
17 | 4 | iscld2 12276 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐴 ∪ 𝐵) ⊆ ∪ 𝐽) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽)) |
18 | 3, 16, 17 | syl2anc 408 | . 2 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ (𝐴 ∪ 𝐵)) ∈ 𝐽)) |
19 | 11, 18 | mpbird 166 | 1 ⊢ ((𝐴 ∈ (Clsd‘𝐽) ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∪ 𝐵) ∈ (Clsd‘𝐽)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 1480 ∖ cdif 3068 ∪ cun 3069 ∩ cin 3070 ⊆ wss 3071 ∪ cuni 3736 ‘cfv 5123 Topctop 12167 Clsdccld 12264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 df-top 12168 df-cld 12267 |
This theorem is referenced by: iuncld 12287 |
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