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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 5420 | . . . . . . . 8 β’ {π₯} β V | |
2 | distop 22842 | . . . . . . . 8 β’ ({π₯} β V β π« {π₯} β Top) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 β’ π« {π₯} β Top |
4 | tgtop 22820 | . . . . . . 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 22819 | . . . . . . . 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 9041 | . . . . . . . . . 10 β’ {π₯} β Fin | |
9 | pwfi 9175 | . . . . . . . . . 10 β’ ({π₯} β Fin β π« {π₯} β Fin) | |
10 | 8, 9 | mpbi 229 | . . . . . . . . 9 β’ π« {π₯} β Fin |
11 | isfinite 9644 | . . . . . . . . 9 β’ (π« {π₯} β Fin β π« {π₯} βΊ Ο) | |
12 | 10, 11 | mpbi 229 | . . . . . . . 8 β’ π« {π₯} βΊ Ο |
13 | sdomdom 8973 | . . . . . . . 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 23296 | . . . . . . 7 β’ ((π« {π₯} β TopBases β§ π« {π₯} βΌ Ο) β (topGenβπ« {π₯}) β 2ndΟ) | |
16 | 7, 14, 15 | mp2an 689 | . . . . . 6 β’ (topGenβπ« {π₯}) β 2ndΟ |
17 | 5, 16 | eqeltrri 2822 | . . . . 5 β’ π« {π₯} β 2ndΟ |
18 | 2ndc1stc 23299 | . . . . 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 3057 | . . 3 β’ βπ₯ β π π« {π₯} β 1stΟ |
21 | dislly 23345 | . . 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 23344 | . 2 β’ Locally 1stΟ = 1stΟ | |
24 | 22, 23 | eleqtrdi 2835 | 1 β’ (π β π β π« π β 1stΟ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwral 3053 Vcvv 3466 π« cpw 4595 {csn 4621 class class class wbr 5139 βcfv 6534 Οcom 7849 βΌ cdom 8934 βΊ csdm 8935 Fincfn 8936 topGenctg 17388 Topctop 22739 TopBasesctb 22792 1stΟc1stc 23285 2ndΟc2ndc 23286 Locally clly 23312 |
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-inf2 9633 |
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-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-op 4628 df-uni 4901 df-int 4942 df-iun 4990 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-se 5623 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-isom 6543 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-card 9931 df-acn 9934 df-rest 17373 df-topgen 17394 df-top 22740 df-topon 22757 df-bases 22793 df-1stc 23287 df-2ndc 23288 df-lly 23314 |
This theorem is referenced by: (None) |
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