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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 4621). (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 21062 | . . 3 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → ({∅, {𝐴}} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 {𝐴})) | |
2 | 1 | simprd 490 | . 2 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 ⊆ 𝒫 {𝐴}) |
3 | pwsn 4621 | . . 3 ⊢ 𝒫 {𝐴} = {∅, {𝐴}} | |
4 | 1 | simpld 489 | . . 3 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → {∅, {𝐴}} ⊆ 𝐽) |
5 | 3, 4 | syl5eqss 3846 | . 2 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝒫 {𝐴} ⊆ 𝐽) |
6 | 2, 5 | eqssd 3816 | 1 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ⊆ wss 3770 ∅c0 4116 𝒫 cpw 4350 {csn 4369 {cpr 4371 ‘cfv 6102 TopOnctopon 21042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-iota 6065 df-fun 6104 df-fv 6110 df-top 21026 df-topon 21043 |
This theorem is referenced by: restsn2 21303 rrxtopn0 41251 |
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