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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 | snex 5324 | . . . . . . . 8 ⊢ {𝑥} ∈ V | |
2 | distop 21892 | . . . . . . . 8 ⊢ ({𝑥} ∈ V → 𝒫 {𝑥} ∈ Top) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ Top |
4 | tgtop 21870 | . . . . . . 7 ⊢ (𝒫 {𝑥} ∈ Top → (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥}) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥} |
6 | topbas 21869 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ∈ Top → 𝒫 {𝑥} ∈ TopBases) | |
7 | 3, 6 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ TopBases |
8 | snfi 8721 | . . . . . . . . . 10 ⊢ {𝑥} ∈ Fin | |
9 | pwfi 8856 | . . . . . . . . . 10 ⊢ ({𝑥} ∈ Fin ↔ 𝒫 {𝑥} ∈ Fin) | |
10 | 8, 9 | mpbi 233 | . . . . . . . . 9 ⊢ 𝒫 {𝑥} ∈ Fin |
11 | isfinite 9267 | . . . . . . . . 9 ⊢ (𝒫 {𝑥} ∈ Fin ↔ 𝒫 {𝑥} ≺ ω) | |
12 | 10, 11 | mpbi 233 | . . . . . . . 8 ⊢ 𝒫 {𝑥} ≺ ω |
13 | sdomdom 8656 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ≺ ω → 𝒫 {𝑥} ≼ ω) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ≼ ω |
15 | 2ndci 22345 | . . . . . . 7 ⊢ ((𝒫 {𝑥} ∈ TopBases ∧ 𝒫 {𝑥} ≼ ω) → (topGen‘𝒫 {𝑥}) ∈ 2ndω) | |
16 | 7, 14, 15 | mp2an 692 | . . . . . 6 ⊢ (topGen‘𝒫 {𝑥}) ∈ 2ndω |
17 | 5, 16 | eqeltrri 2835 | . . . . 5 ⊢ 𝒫 {𝑥} ∈ 2ndω |
18 | 2ndc1stc 22348 | . . . . 5 ⊢ (𝒫 {𝑥} ∈ 2ndω → 𝒫 {𝑥} ∈ 1stω) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ 𝒫 {𝑥} ∈ 1stω |
20 | 19 | rgenw 3073 | . . 3 ⊢ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 1stω |
21 | dislly 22394 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 1stω ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 1stω)) | |
22 | 20, 21 | mpbiri 261 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Locally 1stω) |
23 | lly1stc 22393 | . 2 ⊢ Locally 1stω = 1stω | |
24 | 22, 23 | eleqtrdi 2848 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ 1stω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 𝒫 cpw 4513 {csn 4541 class class class wbr 5053 ‘cfv 6380 ωcom 7644 ≼ cdom 8624 ≺ csdm 8625 Fincfn 8626 topGenctg 16942 Topctop 21790 TopBasesctb 21842 1stωc1stc 22334 2ndωc2ndc 22335 Locally clly 22361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-card 9555 df-acn 9558 df-rest 16927 df-topgen 16948 df-top 21791 df-topon 21808 df-bases 21843 df-1stc 22336 df-2ndc 22337 df-lly 22363 |
This theorem is referenced by: (None) |
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