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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 5431 | . . . . . . . 8 β’ {π₯} β V | |
| 2 | distop 22897 | . . . . . . . 8 β’ ({π₯} β V β π« {π₯} β Top) | |
| 3 | 1, 2 | ax-mp 5 | . . . . . . 7 β’ π« {π₯} β Top |
| 4 | tgtop 22875 | . . . . . . 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 22874 | . . . . . . . 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 9068 | . . . . . . . . . 10 β’ {π₯} β Fin | |
| 9 | pwfi 9202 | . . . . . . . . . 10 β’ ({π₯} β Fin β π« {π₯} β Fin) | |
| 10 | 8, 9 | mpbi 229 | . . . . . . . . 9 β’ π« {π₯} β Fin |
| 11 | isfinite 9675 | . . . . . . . . 9 β’ (π« {π₯} β Fin β π« {π₯} βΊ Ο) | |
| 12 | 10, 11 | mpbi 229 | . . . . . . . 8 β’ π« {π₯} βΊ Ο |
| 13 | sdomdom 9000 | . . . . . . . 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 23351 | . . . . . . 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 2826 | . . . . 5 β’ π« {π₯} β 2ndΟ |
| 18 | 2ndc1stc 23354 | . . . . 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 3062 | . . 3 β’ βπ₯ β π π« {π₯} β 1stΟ |
| 21 | dislly 23400 | . . 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 23399 | . 2 β’ Locally 1stΟ = 1stΟ | |
| 24 | 22, 23 | eleqtrdi 2839 | 1 β’ (π β π β π« π β 1stΟ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1534 β wcel 2099 βwral 3058 Vcvv 3471 π« cpw 4603 {csn 4629 class class class wbr 5148 βcfv 6548 Οcom 7870 βΌ cdom 8961 βΊ csdm 8962 Fincfn 8963 topGenctg 17418 Topctop 22794 TopBasesctb 22847 1stΟc1stc 23340 2ndΟc2ndc 23341 Locally clly 23367 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fi 9434 df-card 9962 df-acn 9965 df-rest 17403 df-topgen 17424 df-top 22795 df-topon 22812 df-bases 22848 df-1stc 23342 df-2ndc 23343 df-lly 23369 |
| This theorem is referenced by: (None) |
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