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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 5334 | . . . . . . . 8 ⊢ {𝑥} ∈ V | |
2 | distop 21605 | . . . . . . . 8 ⊢ ({𝑥} ∈ V → 𝒫 {𝑥} ∈ Top) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ Top |
4 | tgtop 21583 | . . . . . . 7 ⊢ (𝒫 {𝑥} ∈ Top → (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥}) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥} |
6 | topbas 21582 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ∈ Top → 𝒫 {𝑥} ∈ TopBases) | |
7 | 3, 6 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ TopBases |
8 | snfi 8596 | . . . . . . . . . 10 ⊢ {𝑥} ∈ Fin | |
9 | pwfi 8821 | . . . . . . . . . 10 ⊢ ({𝑥} ∈ Fin ↔ 𝒫 {𝑥} ∈ Fin) | |
10 | 8, 9 | mpbi 232 | . . . . . . . . 9 ⊢ 𝒫 {𝑥} ∈ Fin |
11 | isfinite 9117 | . . . . . . . . 9 ⊢ (𝒫 {𝑥} ∈ Fin ↔ 𝒫 {𝑥} ≺ ω) | |
12 | 10, 11 | mpbi 232 | . . . . . . . 8 ⊢ 𝒫 {𝑥} ≺ ω |
13 | sdomdom 8539 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ≺ ω → 𝒫 {𝑥} ≼ ω) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ≼ ω |
15 | 2ndci 22058 | . . . . . . 7 ⊢ ((𝒫 {𝑥} ∈ TopBases ∧ 𝒫 {𝑥} ≼ ω) → (topGen‘𝒫 {𝑥}) ∈ 2ndω) | |
16 | 7, 14, 15 | mp2an 690 | . . . . . 6 ⊢ (topGen‘𝒫 {𝑥}) ∈ 2ndω |
17 | 5, 16 | eqeltrri 2912 | . . . . 5 ⊢ 𝒫 {𝑥} ∈ 2ndω |
18 | 2ndc1stc 22061 | . . . . 5 ⊢ (𝒫 {𝑥} ∈ 2ndω → 𝒫 {𝑥} ∈ 1stω) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ 𝒫 {𝑥} ∈ 1stω |
20 | 19 | rgenw 3152 | . . 3 ⊢ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 1stω |
21 | dislly 22107 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ Locally 1stω ↔ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 1stω)) | |
22 | 20, 21 | mpbiri 260 | . 2 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ Locally 1stω) |
23 | lly1stc 22106 | . 2 ⊢ Locally 1stω = 1stω | |
24 | 22, 23 | eleqtrdi 2925 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ 1stω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 𝒫 cpw 4541 {csn 4569 class class class wbr 5068 ‘cfv 6357 ωcom 7582 ≼ cdom 8509 ≺ csdm 8510 Fincfn 8511 topGenctg 16713 Topctop 21503 TopBasesctb 21555 1stωc1stc 22047 2ndωc2ndc 22048 Locally clly 22074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fi 8877 df-card 9370 df-acn 9373 df-rest 16698 df-topgen 16719 df-top 21504 df-topon 21521 df-bases 21556 df-1stc 22049 df-2ndc 22050 df-lly 22076 |
This theorem is referenced by: (None) |
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