Theorem topsn 22080

Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4831). (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

Step	Hyp	Ref	Expression
1		topgele 22079	. . 3 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → ({∅, {𝐴}} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 {𝐴}))
2	1	simprd 496	. 2 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 ⊆ 𝒫 {𝐴})
3		pwsn 4831	. . 3 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
4	1	simpld 495	. . 3 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → {∅, {𝐴}} ⊆ 𝐽)
5	3, 4	eqsstrid 3969	. 2 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝒫 {𝐴} ⊆ 𝐽)
6	2, 5	eqssd 3938	1 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  {csn 4561  {cpr 4563  ‘cfv 6433  TopOnctopon 22059

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-top 22043  df-topon 22060

This theorem is referenced by:  restsn2  22322  rrxtopn0  43834