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| Mirrors > Home > MPE Home > Th. List > topopn | Structured version Visualization version 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 4006 | . . 3 ⊢ 𝐽 ⊆ 𝐽 | |
| 3 | uniopn 22903 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ⊆ 𝐽) → ∪ 𝐽 ∈ 𝐽) | |
| 4 | 2, 3 | mpan2 691 | . 2 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
| 5 | 1, 4 | eqeltrid 2845 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 Topctop 22899 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-pw 4602 df-uni 4908 df-top 22900 |
| This theorem is referenced by: riinopn 22914 toponmax 22932 cldval 23031 ntrfval 23032 clsfval 23033 iscld 23035 ntrval 23044 clsval 23045 0cld 23046 clsval2 23058 ntrtop 23078 toponmre 23101 neifval 23107 neif 23108 neival 23110 isnei 23111 tpnei 23129 lpfval 23146 lpval 23147 restcld 23180 restcls 23189 restntr 23190 cnrest 23293 cmpsub 23408 hauscmplem 23414 cmpfi 23416 isconn2 23422 connsubclo 23432 1stcfb 23453 1stcelcls 23469 islly2 23492 lly1stc 23504 islocfin 23525 finlocfin 23528 cmpkgen 23559 llycmpkgen 23560 ptbasid 23583 ptpjpre2 23588 ptopn2 23592 xkoopn 23597 xkouni 23607 txcld 23611 txcn 23634 ptrescn 23647 txtube 23648 txhaus 23655 xkoptsub 23662 xkopt 23663 xkopjcn 23664 qtoptop 23708 qtopuni 23710 opnfbas 23850 flimval 23971 flimfil 23977 hausflim 23989 hauspwpwf1 23995 hauspwpwdom 23996 flimfnfcls 24036 cnpfcfi 24048 bcthlem5 25362 dvply1 26325 cldssbrsiga 34188 dya2iocucvr 34286 kur14lem7 35217 kur14lem9 35219 connpconn 35240 cvmliftmolem1 35286 ordtop 36437 pibt2 37418 ntrelmap 44138 clselmap 44140 dssmapntrcls 44141 dssmapclsntr 44142 toprestsubel 45159 reopn 45301 toplatglb0 48888 |
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