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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 21892 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) | |
2 | unipw 5335 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 2 | eqcomi 2746 | . 2 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
4 | istopon 21809 | . 2 ⊢ (𝒫 𝐴 ∈ (TopOn‘𝐴) ↔ (𝒫 𝐴 ∈ Top ∧ 𝐴 = ∪ 𝒫 𝐴)) | |
5 | 1, 3, 4 | sylanblrc 593 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 𝒫 cpw 4513 ∪ cuni 4819 ‘cfv 6380 Topctop 21790 TopOnctopon 21807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fv 6388 df-top 21791 df-topon 21808 |
This theorem is referenced by: sn0topon 21895 toponmre 21990 cndis 22188 txdis1cn 22532 xkofvcn 22581 distgp 22996 efmndtmd 22998 symgtgp 23003 cnfdmsn 43098 |
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