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Mirrors > Home > MPE Home > Th. List > sszcld | Structured version Visualization version GIF version |
Description: Every subset of the integers are closed in the topology on β. (Contributed by Mario Carneiro, 6-Jul-2017.) |
Ref | Expression |
---|---|
recld2.1 | β’ π½ = (TopOpenββfld) |
Ref | Expression |
---|---|
sszcld | β’ (π΄ β β€ β π΄ β (Clsdβπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recld2.1 | . . 3 β’ π½ = (TopOpenββfld) | |
2 | 1 | zcld2 24675 | . 2 β’ β€ β (Clsdβπ½) |
3 | id 22 | . . 3 β’ (π΄ β β€ β π΄ β β€) | |
4 | zex 12566 | . . . . 5 β’ β€ β V | |
5 | difss 4124 | . . . . 5 β’ (β€ β π΄) β β€ | |
6 | 4, 5 | elpwi2 5337 | . . . 4 β’ (β€ β π΄) β π« β€ |
7 | 1 | zdis 24676 | . . . 4 β’ (π½ βΎt β€) = π« β€ |
8 | 6, 7 | eleqtrri 2824 | . . 3 β’ (β€ β π΄) β (π½ βΎt β€) |
9 | 1 | cnfldtopon 24643 | . . . . . 6 β’ π½ β (TopOnββ) |
10 | zsscn 12565 | . . . . . 6 β’ β€ β β | |
11 | resttopon 23009 | . . . . . 6 β’ ((π½ β (TopOnββ) β§ β€ β β) β (π½ βΎt β€) β (TopOnββ€)) | |
12 | 9, 10, 11 | mp2an 689 | . . . . 5 β’ (π½ βΎt β€) β (TopOnββ€) |
13 | 12 | topontopi 22761 | . . . 4 β’ (π½ βΎt β€) β Top |
14 | 12 | toponunii 22762 | . . . . 5 β’ β€ = βͺ (π½ βΎt β€) |
15 | 14 | iscld 22875 | . . . 4 β’ ((π½ βΎt β€) β Top β (π΄ β (Clsdβ(π½ βΎt β€)) β (π΄ β β€ β§ (β€ β π΄) β (π½ βΎt β€)))) |
16 | 13, 15 | ax-mp 5 | . . 3 β’ (π΄ β (Clsdβ(π½ βΎt β€)) β (π΄ β β€ β§ (β€ β π΄) β (π½ βΎt β€))) |
17 | 3, 8, 16 | sylanblrc 589 | . 2 β’ (π΄ β β€ β π΄ β (Clsdβ(π½ βΎt β€))) |
18 | restcldr 23022 | . 2 β’ ((β€ β (Clsdβπ½) β§ π΄ β (Clsdβ(π½ βΎt β€))) β π΄ β (Clsdβπ½)) | |
19 | 2, 17, 18 | sylancr 586 | 1 β’ (π΄ β β€ β π΄ β (Clsdβπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3466 β cdif 3938 β wss 3941 π« cpw 4595 βcfv 6534 (class class class)co 7402 βcc 11105 β€cz 12557 βΎt crest 17371 TopOpenctopn 17372 βfldccnfld 21234 Topctop 22739 TopOnctopon 22756 Clsdccld 22864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-ioo 13329 df-fz 13486 df-fl 13758 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-rest 17373 df-topn 17374 df-topgen 17394 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-cnfld 21235 df-top 22740 df-topon 22757 df-topsp 22779 df-bases 22793 df-cld 22867 df-xms 24170 df-ms 24171 |
This theorem is referenced by: lgamucov 26911 |
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