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Mirrors > Home > ILE Home > Th. List > distps | GIF version |
Description: The discrete topology on a set 𝐴 expressed as a topological space. (Contributed by FL, 20-Aug-2006.) |
Ref | Expression |
---|---|
distps.a | ⊢ 𝐴 ∈ V |
distps.k | ⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝒫 𝐴〉} |
Ref | Expression |
---|---|
distps | ⊢ 𝐾 ∈ TopSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distps.k | . 2 ⊢ 𝐾 = {〈(Base‘ndx), 𝐴〉, 〈(TopSet‘ndx), 𝒫 𝐴〉} | |
2 | unipw 4134 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 2 | eqcomi 2141 | . 2 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
4 | distps.a | . . 3 ⊢ 𝐴 ∈ V | |
5 | distop 12243 | . . 3 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ Top) | |
6 | 4, 5 | ax-mp 5 | . 2 ⊢ 𝒫 𝐴 ∈ Top |
7 | 1, 3, 6 | eltpsi 12197 | 1 ⊢ 𝐾 ∈ TopSp |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2681 𝒫 cpw 3505 {cpr 3523 〈cop 3525 ∪ cuni 3731 ‘cfv 5118 ndxcnx 11945 Basecbs 11948 TopSetcts 12016 Topctop 12153 TopSpctps 12186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-pre-ltirr 7725 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-5 8775 df-6 8776 df-7 8777 df-8 8778 df-9 8779 df-ndx 11951 df-slot 11952 df-base 11954 df-tset 12029 df-rest 12111 df-topn 12112 df-top 12154 df-topon 12167 df-topsp 12187 |
This theorem is referenced by: (None) |
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