Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cldss | GIF version |
Description: A closed set is a subset of the underlying set of a topology. (Contributed by NM, 5-Oct-2006.) (Revised by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
cldss | ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cldrcl 12198 | . 2 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | |
2 | iscld.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | iscld 12199 | . . 3 ⊢ (𝐽 ∈ Top → (𝑆 ∈ (Clsd‘𝐽) ↔ (𝑆 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑆) ∈ 𝐽))) |
4 | 3 | simprbda 380 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ (Clsd‘𝐽)) → 𝑆 ⊆ 𝑋) |
5 | 1, 4 | mpancom 418 | 1 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ∖ cdif 3038 ⊆ wss 3041 ∪ cuni 3706 ‘cfv 5093 Topctop 12091 Clsdccld 12188 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 df-top 12092 df-cld 12191 |
This theorem is referenced by: cldss2 12202 uncld 12209 cldcls 12210 clsss2 12225 |
Copyright terms: Public domain | W3C validator |