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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sncldre | Structured version Visualization version GIF version |
Description: A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
sncldre | ⊢ (𝐴 ∈ ℝ → {𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rehaus 22821 | . 2 ⊢ (topGen‘ran (,)) ∈ Haus | |
2 | uniretop 22785 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
3 | 2 | sncld 21395 | . 2 ⊢ (((topGen‘ran (,)) ∈ Haus ∧ 𝐴 ∈ ℝ) → {𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
4 | 1, 3 | mpan 662 | 1 ⊢ (𝐴 ∈ ℝ → {𝐴} ∈ (Clsd‘(topGen‘ran (,)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2145 {csn 4316 ran crn 5250 ‘cfv 6031 ℝcr 10136 (,)cioo 12379 topGenctg 16305 Clsdccld 21040 Hauscha 21332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-sup 8503 df-inf 8504 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-n0 11494 df-z 11579 df-uz 11888 df-q 11991 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-ioo 12383 df-icc 12386 df-seq 13008 df-exp 13067 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-topgen 16311 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-top 20918 df-topon 20935 df-bases 20970 df-cld 21043 df-t1 21338 df-haus 21339 |
This theorem is referenced by: fourierdlem62 40898 |
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