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Theorem topsn 21632
 Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4791). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
topsn (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴})

Proof of Theorem topsn
StepHypRef Expression
1 topgele 21631 . . 3 (𝐽 ∈ (TopOn‘{𝐴}) → ({∅, {𝐴}} ⊆ 𝐽𝐽 ⊆ 𝒫 {𝐴}))
21simprd 500 . 2 (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 ⊆ 𝒫 {𝐴})
3 pwsn 4791 . . 3 𝒫 {𝐴} = {∅, {𝐴}}
41simpld 499 . . 3 (𝐽 ∈ (TopOn‘{𝐴}) → {∅, {𝐴}} ⊆ 𝐽)
53, 4eqsstrid 3941 . 2 (𝐽 ∈ (TopOn‘{𝐴}) → 𝒫 {𝐴} ⊆ 𝐽)
62, 5eqssd 3910 1 (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112   ⊆ wss 3859  ∅c0 4226  𝒫 cpw 4495  {csn 4523  {cpr 4525  ‘cfv 6336  TopOnctopon 21611 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6295  df-fun 6338  df-fv 6344  df-top 21595  df-topon 21612 This theorem is referenced by:  restsn2  21872  rrxtopn0  43302
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