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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 4018 | . . 3 ⊢ 𝐽 ⊆ 𝐽 | |
3 | uniopn 22919 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ⊆ 𝐽) → ∪ 𝐽 ∈ 𝐽) | |
4 | 2, 3 | mpan2 691 | . 2 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 1, 4 | eqeltrid 2843 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∪ cuni 4912 Topctop 22915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-in 3970 df-ss 3980 df-pw 4607 df-uni 4913 df-top 22916 |
This theorem is referenced by: riinopn 22930 toponmax 22948 cldval 23047 ntrfval 23048 clsfval 23049 iscld 23051 ntrval 23060 clsval 23061 0cld 23062 clsval2 23074 ntrtop 23094 toponmre 23117 neifval 23123 neif 23124 neival 23126 isnei 23127 tpnei 23145 lpfval 23162 lpval 23163 restcld 23196 restcls 23205 restntr 23206 cnrest 23309 cmpsub 23424 hauscmplem 23430 cmpfi 23432 isconn2 23438 connsubclo 23448 1stcfb 23469 1stcelcls 23485 islly2 23508 lly1stc 23520 islocfin 23541 finlocfin 23544 cmpkgen 23575 llycmpkgen 23576 ptbasid 23599 ptpjpre2 23604 ptopn2 23608 xkoopn 23613 xkouni 23623 txcld 23627 txcn 23650 ptrescn 23663 txtube 23664 txhaus 23671 xkoptsub 23678 xkopt 23679 xkopjcn 23680 qtoptop 23724 qtopuni 23726 opnfbas 23866 flimval 23987 flimfil 23993 hausflim 24005 hauspwpwf1 24011 hauspwpwdom 24012 flimfnfcls 24052 cnpfcfi 24064 bcthlem5 25376 dvply1 26340 cldssbrsiga 34168 dya2iocucvr 34266 kur14lem7 35197 kur14lem9 35199 connpconn 35220 cvmliftmolem1 35266 ordtop 36419 pibt2 37400 ntrelmap 44115 clselmap 44117 dssmapntrcls 44118 dssmapclsntr 44119 toprestsubel 45097 reopn 45240 toplatglb0 48788 |
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