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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 5429 | . . . . . . . 8 β’ {π₯} β V | |
| 2 | distop 22497 | . . . . . . . 8 β’ ({π₯} β V β π« {π₯} β Top) | |
| 3 | 1, 2 | ax-mp 5 | . . . . . . 7 β’ π« {π₯} β Top |
| 4 | tgtop 22475 | . . . . . . 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 22474 | . . . . . . . 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 9043 | . . . . . . . . . 10 β’ {π₯} β Fin | |
| 9 | pwfi 9177 | . . . . . . . . . 10 β’ ({π₯} β Fin β π« {π₯} β Fin) | |
| 10 | 8, 9 | mpbi 229 | . . . . . . . . 9 β’ π« {π₯} β Fin |
| 11 | isfinite 9646 | . . . . . . . . 9 β’ (π« {π₯} β Fin β π« {π₯} βΊ Ο) | |
| 12 | 10, 11 | mpbi 229 | . . . . . . . 8 β’ π« {π₯} βΊ Ο |
| 13 | sdomdom 8975 | . . . . . . . 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 22951 | . . . . . . 7 β’ ((π« {π₯} β TopBases β§ π« {π₯} βΌ Ο) β (topGenβπ« {π₯}) β 2ndΟ) | |
| 16 | 7, 14, 15 | mp2an 690 | . . . . . 6 β’ (topGenβπ« {π₯}) β 2ndΟ |
| 17 | 5, 16 | eqeltrri 2830 | . . . . 5 β’ π« {π₯} β 2ndΟ |
| 18 | 2ndc1stc 22954 | . . . . 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 23000 | . . 3 β’ (π β π β (π« π β Locally 1stΟ β βπ₯ β π π« {π₯} β 1stΟ)) | |
| 22 | 20, 21 | mpbiri 257 | . 2 β’ (π β π β π« π β Locally 1stΟ) |
| 23 | lly1stc 22999 | . 2 β’ Locally 1stΟ = 1stΟ | |
| 24 | 22, 23 | eleqtrdi 2843 | 1 β’ (π β π β π« π β 1stΟ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 = wceq 1541 β wcel 2106 βwral 3061 Vcvv 3474 π« cpw 4602 {csn 4628 class class class wbr 5148 βcfv 6543 Οcom 7854 βΌ cdom 8936 βΊ csdm 8937 Fincfn 8938 topGenctg 17382 Topctop 22394 TopBasesctb 22447 1stΟc1stc 22940 2ndΟc2ndc 22941 Locally clly 22967 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-card 9933 df-acn 9936 df-rest 17367 df-topgen 17388 df-top 22395 df-topon 22412 df-bases 22448 df-1stc 22942 df-2ndc 22943 df-lly 22969 |
| This theorem is referenced by: (None) |
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