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Theorem topsn 22757
Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4893). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
topsn (𝐽 ∈ (TopOnβ€˜{𝐴}) β†’ 𝐽 = 𝒫 {𝐴})

Proof of Theorem topsn
StepHypRef Expression
1 topgele 22756 . . 3 (𝐽 ∈ (TopOnβ€˜{𝐴}) β†’ ({βˆ…, {𝐴}} βŠ† 𝐽 ∧ 𝐽 βŠ† 𝒫 {𝐴}))
21simprd 495 . 2 (𝐽 ∈ (TopOnβ€˜{𝐴}) β†’ 𝐽 βŠ† 𝒫 {𝐴})
3 pwsn 4893 . . 3 𝒫 {𝐴} = {βˆ…, {𝐴}}
41simpld 494 . . 3 (𝐽 ∈ (TopOnβ€˜{𝐴}) β†’ {βˆ…, {𝐴}} βŠ† 𝐽)
53, 4eqsstrid 4023 . 2 (𝐽 ∈ (TopOnβ€˜{𝐴}) β†’ 𝒫 {𝐴} βŠ† 𝐽)
62, 5eqssd 3992 1 (𝐽 ∈ (TopOnβ€˜{𝐴}) β†’ 𝐽 = 𝒫 {𝐴})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   βŠ† wss 3941  βˆ…c0 4315  π’« cpw 4595  {csn 4621  {cpr 4623  β€˜cfv 6534  TopOnctopon 22736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-top 22720  df-topon 22737
This theorem is referenced by:  restsn2  22999  rrxtopn0  45519
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