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| Mirrors > Home > ILE Home > Th. List > topcld | GIF version | ||
| Description: The underlying set of a topology is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 3-Oct-2006.) |
| Ref | Expression |
|---|---|
| iscld.1 | ⊢ 𝑋 = ∪ 𝐽 |
| Ref | Expression |
|---|---|
| topcld | ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | difid 3528 | . . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅ | |
| 2 | 0opn 14449 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
| 3 | 1, 2 | eqeltrid 2291 | . . 3 ⊢ (𝐽 ∈ Top → (𝑋 ∖ 𝑋) ∈ 𝐽) |
| 4 | ssid 3212 | . . 3 ⊢ 𝑋 ⊆ 𝑋 | |
| 5 | 3, 4 | jctil 312 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽)) |
| 6 | iscld.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
| 7 | 6 | iscld 14546 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ∈ (Clsd‘𝐽) ↔ (𝑋 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑋) ∈ 𝐽))) |
| 8 | 5, 7 | mpbird 167 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 = wceq 1372 ∈ wcel 2175 ∖ cdif 3162 ⊆ wss 3165 ∅c0 3459 ∪ cuni 3849 ‘cfv 5270 Topctop 14440 Clsdccld 14535 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-rab 2492 df-v 2773 df-sbc 2998 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-iota 5231 df-fun 5272 df-fv 5278 df-top 14441 df-cld 14538 |
| This theorem is referenced by: clsval 14554 clstop 14570 clsss3 14573 |
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