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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 5230 | . . . . . . . 8 ⊢ {𝑥} ∈ V | |
2 | distop 21291 | . . . . . . . 8 ⊢ ({𝑥} ∈ V → 𝒫 {𝑥} ∈ Top) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ Top |
4 | tgtop 21269 | . . . . . . 7 ⊢ (𝒫 {𝑥} ∈ Top → (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥}) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥} |
6 | topbas 21268 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ∈ Top → 𝒫 {𝑥} ∈ TopBases) | |
7 | 3, 6 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ TopBases |
8 | snfi 8449 | . . . . . . . . . 10 ⊢ {𝑥} ∈ Fin | |
9 | pwfi 8672 | . . . . . . . . . 10 ⊢ ({𝑥} ∈ Fin ↔ 𝒫 {𝑥} ∈ Fin) | |
10 | 8, 9 | mpbi 231 | . . . . . . . . 9 ⊢ 𝒫 {𝑥} ∈ Fin |
11 | isfinite 8968 | . . . . . . . . 9 ⊢ (𝒫 {𝑥} ∈ Fin ↔ 𝒫 {𝑥} ≺ ω) | |
12 | 10, 11 | mpbi 231 | . . . . . . . 8 ⊢ 𝒫 {𝑥} ≺ ω |
13 | sdomdom 8392 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ≺ ω → 𝒫 {𝑥} ≼ ω) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ≼ ω |
15 | 2ndci 21744 | . . . . . . 7 ⊢ ((𝒫 {𝑥} ∈ TopBases ∧ 𝒫 {𝑥} ≼ ω) → (topGen‘𝒫 {𝑥}) ∈ 2nd𝜔) | |
16 | 7, 14, 15 | mp2an 688 | . . . . . 6 ⊢ (topGen‘𝒫 {𝑥}) ∈ 2nd𝜔 |
17 | 5, 16 | eqeltrri 2882 | . . . . 5 ⊢ 𝒫 {𝑥} ∈ 2nd𝜔 |
18 | 2ndc1stc 21747 | . . . . 5 ⊢ (𝒫 {𝑥} ∈ 2nd𝜔 → 𝒫 {𝑥} ∈ 1st𝜔) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ 𝒫 {𝑥} ∈ 1st𝜔 |
20 | 19 | rgenw 3119 | . . 3 ⊢ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 1st𝜔 |
21 | dislly 21793 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 1st𝜔 ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 1st𝜔)) | |
22 | 20, 21 | mpbiri 259 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Locally 1st𝜔) |
23 | lly1stc 21792 | . 2 ⊢ Locally 1st𝜔 = 1st𝜔 | |
24 | 22, 23 | syl6eleq 2895 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ 1st𝜔) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∈ wcel 2083 ∀wral 3107 Vcvv 3440 𝒫 cpw 4459 {csn 4478 class class class wbr 4968 ‘cfv 6232 ωcom 7443 ≼ cdom 8362 ≺ csdm 8363 Fincfn 8364 topGenctg 16544 Topctop 21189 TopBasesctb 21241 1st𝜔c1stc 21733 2nd𝜔c2ndc 21734 Locally clly 21760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-inf2 8957 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-2o 7961 df-oadd 7964 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fi 8728 df-card 9221 df-acn 9224 df-rest 16529 df-topgen 16550 df-top 21190 df-topon 21207 df-bases 21242 df-1stc 21735 df-2ndc 21736 df-lly 21762 |
This theorem is referenced by: (None) |
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