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Mirrors > Home > MPE Home > Th. List > distopon | Structured version Visualization version GIF version |
Description: The discrete topology on a set 𝐴, with base set. (Contributed by Mario Carneiro, 13-Aug-2015.) |
Ref | Expression |
---|---|
distopon | ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distop 22981 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) | |
2 | unipw 5455 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 2 | eqcomi 2734 | . 2 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
4 | istopon 22897 | . 2 ⊢ (𝒫 𝐴 ∈ (TopOn‘𝐴) ↔ (𝒫 𝐴 ∈ Top ∧ 𝐴 = ∪ 𝒫 𝐴)) | |
5 | 1, 3, 4 | sylanblrc 588 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 𝒫 cpw 4606 ∪ cuni 4912 ‘cfv 6553 Topctop 22878 TopOnctopon 22895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-iota 6505 df-fun 6555 df-fv 6561 df-top 22879 df-topon 22896 |
This theorem is referenced by: sn0topon 22984 toponmre 23080 cndis 23278 txdis1cn 23622 xkofvcn 23671 distgp 24086 efmndtmd 24088 symgtgp 24093 cnfdmsn 45440 |
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