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Mirrors > Home > MPE Home > Th. List > topsn | Structured version Visualization version GIF version |
Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4855). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
topsn | ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | topgele 22231 | . . 3 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → ({∅, {𝐴}} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 {𝐴})) | |
2 | 1 | simprd 496 | . 2 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 ⊆ 𝒫 {𝐴}) |
3 | pwsn 4855 | . . 3 ⊢ 𝒫 {𝐴} = {∅, {𝐴}} | |
4 | 1 | simpld 495 | . . 3 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → {∅, {𝐴}} ⊆ 𝐽) |
5 | 3, 4 | eqsstrid 3990 | . 2 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝒫 {𝐴} ⊆ 𝐽) |
6 | 2, 5 | eqssd 3959 | 1 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 ∅c0 4280 𝒫 cpw 4558 {csn 4584 {cpr 4586 ‘cfv 6493 TopOnctopon 22211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-top 22195 df-topon 22212 |
This theorem is referenced by: restsn2 22474 rrxtopn0 44435 |
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