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Mirrors > Home > MPE Home > Th. List > dis1stc | Structured version Visualization version GIF version |
Description: A discrete space is first-countable. (Contributed by Mario Carneiro, 21-Mar-2015.) |
Ref | Expression |
---|---|
dis1stc | β’ (π β π β π« π β 1stΟ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vsnex 5390 | . . . . . . . 8 β’ {π₯} β V | |
2 | distop 22368 | . . . . . . . 8 β’ ({π₯} β V β π« {π₯} β Top) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 β’ π« {π₯} β Top |
4 | tgtop 22346 | . . . . . . 7 β’ (π« {π₯} β Top β (topGenβπ« {π₯}) = π« {π₯}) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 β’ (topGenβπ« {π₯}) = π« {π₯} |
6 | topbas 22345 | . . . . . . . 8 β’ (π« {π₯} β Top β π« {π₯} β TopBases) | |
7 | 3, 6 | ax-mp 5 | . . . . . . 7 β’ π« {π₯} β TopBases |
8 | snfi 8994 | . . . . . . . . . 10 β’ {π₯} β Fin | |
9 | pwfi 9128 | . . . . . . . . . 10 β’ ({π₯} β Fin β π« {π₯} β Fin) | |
10 | 8, 9 | mpbi 229 | . . . . . . . . 9 β’ π« {π₯} β Fin |
11 | isfinite 9596 | . . . . . . . . 9 β’ (π« {π₯} β Fin β π« {π₯} βΊ Ο) | |
12 | 10, 11 | mpbi 229 | . . . . . . . 8 β’ π« {π₯} βΊ Ο |
13 | sdomdom 8926 | . . . . . . . 8 β’ (π« {π₯} βΊ Ο β π« {π₯} βΌ Ο) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 β’ π« {π₯} βΌ Ο |
15 | 2ndci 22822 | . . . . . . 7 β’ ((π« {π₯} β TopBases β§ π« {π₯} βΌ Ο) β (topGenβπ« {π₯}) β 2ndΟ) | |
16 | 7, 14, 15 | mp2an 691 | . . . . . 6 β’ (topGenβπ« {π₯}) β 2ndΟ |
17 | 5, 16 | eqeltrri 2831 | . . . . 5 β’ π« {π₯} β 2ndΟ |
18 | 2ndc1stc 22825 | . . . . 5 β’ (π« {π₯} β 2ndΟ β π« {π₯} β 1stΟ) | |
19 | 17, 18 | ax-mp 5 | . . . 4 β’ π« {π₯} β 1stΟ |
20 | 19 | rgenw 3065 | . . 3 β’ βπ₯ β π π« {π₯} β 1stΟ |
21 | dislly 22871 | . . 3 β’ (π β π β (π« π β Locally 1stΟ β βπ₯ β π π« {π₯} β 1stΟ)) | |
22 | 20, 21 | mpbiri 258 | . 2 β’ (π β π β π« π β Locally 1stΟ) |
23 | lly1stc 22870 | . 2 β’ Locally 1stΟ = 1stΟ | |
24 | 22, 23 | eleqtrdi 2844 | 1 β’ (π β π β π« π β 1stΟ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βwral 3061 Vcvv 3447 π« cpw 4564 {csn 4590 class class class wbr 5109 βcfv 6500 Οcom 7806 βΌ cdom 8887 βΊ csdm 8888 Fincfn 8889 topGenctg 17327 Topctop 22265 TopBasesctb 22318 1stΟc1stc 22811 2ndΟc2ndc 22812 Locally clly 22838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fi 9355 df-card 9883 df-acn 9886 df-rest 17312 df-topgen 17333 df-top 22266 df-topon 22283 df-bases 22319 df-1stc 22813 df-2ndc 22814 df-lly 22840 |
This theorem is referenced by: (None) |
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