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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 | ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ 1stω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vsnex 5429 | . . . . . . . 8 ⊢ {𝑥} ∈ V | |
2 | distop 22490 | . . . . . . . 8 ⊢ ({𝑥} ∈ V → 𝒫 {𝑥} ∈ Top) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ Top |
4 | tgtop 22468 | . . . . . . 7 ⊢ (𝒫 {𝑥} ∈ Top → (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥}) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥} |
6 | topbas 22467 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ∈ Top → 𝒫 {𝑥} ∈ TopBases) | |
7 | 3, 6 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ TopBases |
8 | snfi 9041 | . . . . . . . . . 10 ⊢ {𝑥} ∈ Fin | |
9 | pwfi 9175 | . . . . . . . . . 10 ⊢ ({𝑥} ∈ Fin ↔ 𝒫 {𝑥} ∈ Fin) | |
10 | 8, 9 | mpbi 229 | . . . . . . . . 9 ⊢ 𝒫 {𝑥} ∈ Fin |
11 | isfinite 9644 | . . . . . . . . 9 ⊢ (𝒫 {𝑥} ∈ Fin ↔ 𝒫 {𝑥} ≺ ω) | |
12 | 10, 11 | mpbi 229 | . . . . . . . 8 ⊢ 𝒫 {𝑥} ≺ ω |
13 | sdomdom 8973 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ≺ ω → 𝒫 {𝑥} ≼ ω) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ≼ ω |
15 | 2ndci 22944 | . . . . . . 7 ⊢ ((𝒫 {𝑥} ∈ TopBases ∧ 𝒫 {𝑥} ≼ ω) → (topGen‘𝒫 {𝑥}) ∈ 2ndω) | |
16 | 7, 14, 15 | mp2an 691 | . . . . . 6 ⊢ (topGen‘𝒫 {𝑥}) ∈ 2ndω |
17 | 5, 16 | eqeltrri 2831 | . . . . 5 ⊢ 𝒫 {𝑥} ∈ 2ndω |
18 | 2ndc1stc 22947 | . . . . 5 ⊢ (𝒫 {𝑥} ∈ 2ndω → 𝒫 {𝑥} ∈ 1stω) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ 𝒫 {𝑥} ∈ 1stω |
20 | 19 | rgenw 3066 | . . 3 ⊢ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 1stω |
21 | dislly 22993 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 1stω ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 1stω)) | |
22 | 20, 21 | mpbiri 258 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Locally 1stω) |
23 | lly1stc 22992 | . 2 ⊢ Locally 1stω = 1stω | |
24 | 22, 23 | eleqtrdi 2844 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ 1stω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 𝒫 cpw 4602 {csn 4628 class class class wbr 5148 ‘cfv 6541 ωcom 7852 ≼ cdom 8934 ≺ csdm 8935 Fincfn 8936 topGenctg 17380 Topctop 22387 TopBasesctb 22440 1stωc1stc 22933 2ndωc2ndc 22934 Locally clly 22960 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 |
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 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 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 17365 df-topgen 17386 df-top 22388 df-topon 22405 df-bases 22441 df-1stc 22935 df-2ndc 22936 df-lly 22962 |
This theorem is referenced by: (None) |
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