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Mirrors > Home > ILE Home > Th. List > topopn | GIF version |
Description: The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
Ref | Expression |
---|---|
1open.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
topopn | ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1open.1 | . 2 ⊢ 𝑋 = ∪ 𝐽 | |
2 | ssid 3122 | . . 3 ⊢ 𝐽 ⊆ 𝐽 | |
3 | uniopn 12207 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ⊆ 𝐽) → ∪ 𝐽 ∈ 𝐽) | |
4 | 2, 3 | mpan2 422 | . 2 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 1, 4 | eqeltrid 2227 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 ⊆ wss 3076 ∪ cuni 3744 Topctop 12203 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-in 3082 df-ss 3089 df-pw 3517 df-uni 3745 df-top 12204 |
This theorem is referenced by: toponmax 12231 cldval 12307 ntrfval 12308 clsfval 12309 iscld 12311 ntrval 12318 clsval 12319 0cld 12320 ntrtop 12336 neifval 12348 neif 12349 neival 12351 isnei 12352 tpnei 12368 cnrest 12443 txcn 12483 |
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