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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 3529 | . . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅ | |
| 2 | 0opn 14478 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽) | |
| 3 | 1, 2 | eqeltrid 2292 | . . 3 ⊢ (𝐽 ∈ Top → (𝑋 ∖ 𝑋) ∈ 𝐽) |
| 4 | ssid 3213 | . . 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 14575 | . 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 1373 ∈ wcel 2176 ∖ cdif 3163 ⊆ wss 3166 ∅c0 3460 ∪ cuni 3850 ‘cfv 5271 Topctop 14469 Clsdccld 14564 |
| 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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 df-top 14470 df-cld 14567 |
| This theorem is referenced by: clsval 14583 clstop 14599 clsss3 14602 |
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