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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 5332 | . . . . . . . 8 ⊢ {𝑥} ∈ V | |
2 | distop 21603 | . . . . . . . 8 ⊢ ({𝑥} ∈ V → 𝒫 {𝑥} ∈ Top) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ Top |
4 | tgtop 21581 | . . . . . . 7 ⊢ (𝒫 {𝑥} ∈ Top → (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥}) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ (topGen‘𝒫 {𝑥}) = 𝒫 {𝑥} |
6 | topbas 21580 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ∈ Top → 𝒫 {𝑥} ∈ TopBases) | |
7 | 3, 6 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ∈ TopBases |
8 | snfi 8594 | . . . . . . . . . 10 ⊢ {𝑥} ∈ Fin | |
9 | pwfi 8819 | . . . . . . . . . 10 ⊢ ({𝑥} ∈ Fin ↔ 𝒫 {𝑥} ∈ Fin) | |
10 | 8, 9 | mpbi 232 | . . . . . . . . 9 ⊢ 𝒫 {𝑥} ∈ Fin |
11 | isfinite 9115 | . . . . . . . . 9 ⊢ (𝒫 {𝑥} ∈ Fin ↔ 𝒫 {𝑥} ≺ ω) | |
12 | 10, 11 | mpbi 232 | . . . . . . . 8 ⊢ 𝒫 {𝑥} ≺ ω |
13 | sdomdom 8537 | . . . . . . . 8 ⊢ (𝒫 {𝑥} ≺ ω → 𝒫 {𝑥} ≼ ω) | |
14 | 12, 13 | ax-mp 5 | . . . . . . 7 ⊢ 𝒫 {𝑥} ≼ ω |
15 | 2ndci 22056 | . . . . . . 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 2910 | . . . . 5 ⊢ 𝒫 {𝑥} ∈ 2ndω |
18 | 2ndc1stc 22059 | . . . . 5 ⊢ (𝒫 {𝑥} ∈ 2ndω → 𝒫 {𝑥} ∈ 1stω) | |
19 | 17, 18 | ax-mp 5 | . . . 4 ⊢ 𝒫 {𝑥} ∈ 1stω |
20 | 19 | rgenw 3150 | . . 3 ⊢ ∀𝑥 ∈ 𝑋 𝒫 {𝑥} ∈ 1stω |
21 | dislly 22105 | . . 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 22104 | . 2 ⊢ Locally 1stω = 1stω | |
24 | 22, 23 | eleqtrdi 2923 | 1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ 1stω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 𝒫 cpw 4539 {csn 4567 class class class wbr 5066 ‘cfv 6355 ωcom 7580 ≼ cdom 8507 ≺ csdm 8508 Fincfn 8509 topGenctg 16711 Topctop 21501 TopBasesctb 21553 1stωc1stc 22045 2ndωc2ndc 22046 Locally clly 22072 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-card 9368 df-acn 9371 df-rest 16696 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 df-1stc 22047 df-2ndc 22048 df-lly 22074 |
This theorem is referenced by: (None) |
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