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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 21307 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top) | |
2 | unipw 5199 | . . 3 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
3 | 2 | eqcomi 2787 | . 2 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
4 | istopon 21224 | . 2 ⊢ (𝒫 𝐴 ∈ (TopOn‘𝐴) ↔ (𝒫 𝐴 ∈ Top ∧ 𝐴 = ∪ 𝒫 𝐴)) | |
5 | 1, 3, 4 | sylanblrc 581 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ (TopOn‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 𝒫 cpw 4422 ∪ cuni 4712 ‘cfv 6188 Topctop 21205 TopOnctopon 21222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 df-top 21206 df-topon 21223 |
This theorem is referenced by: sn0topon 21310 toponmre 21405 cndis 21603 txdis1cn 21947 xkofvcn 21996 distgp 22411 symgtgp 22413 cnfdmsn 41601 |
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