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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 3587 | . . 3 ⊢ 𝐽 ⊆ 𝐽 | |
3 | uniopn 20527 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ⊆ 𝐽) → ∪ 𝐽 ∈ 𝐽) | |
4 | 2, 3 | mpan2 703 | . 2 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ 𝐽) |
5 | 1, 4 | syl5eqel 2692 | 1 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 Topctop 20517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-pw 4110 df-uni 4373 df-top 20521 |
This theorem is referenced by: riinopn 20538 toponmax 20543 cldval 20637 ntrfval 20638 clsfval 20639 iscld 20641 ntrval 20650 clsval 20651 0cld 20652 clsval2 20664 ntrtop 20684 toponmre 20707 neifval 20713 neif 20714 neival 20716 isnei 20717 tpnei 20735 lpfval 20752 lpval 20753 restcld 20786 restcls 20795 restntr 20796 cnrest 20899 cmpsub 21013 hauscmplem 21019 cmpfi 21021 iscon2 21027 consubclo 21037 1stcfb 21058 1stcelcls 21074 islly2 21097 lly1stc 21109 islocfin 21130 finlocfin 21133 cmpkgen 21164 llycmpkgen 21165 ptbasid 21188 ptpjpre2 21193 ptopn2 21197 xkoopn 21202 xkouni 21212 txcld 21216 txcn 21239 ptrescn 21252 txtube 21253 txhaus 21260 xkoptsub 21267 xkopt 21268 xkopjcn 21269 qtoptop 21313 qtopuni 21315 opnfbas 21456 flimval 21577 flimfil 21583 hausflim 21595 hauspwpwf1 21601 hauspwpwdom 21602 flimfnfcls 21642 cnpfcfi 21654 bcthlem5 22933 dvply1 23843 cldssbrsiga 29577 dya2iocucvr 29673 kur14lem7 30448 kur14lem9 30450 conpcon 30471 cvmliftmolem1 30517 ordtop 31605 ntrelmap 37443 clselmap 37445 dssmapntrcls 37446 dssmapclsntr 37447 reopn 38442 |
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