Theorem topsn 22914

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 22913	. . 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 3953	. 2 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝒫 {𝐴} ⊆ 𝐽)
6	2, 5	eqssd 3932	1 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴})

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  {csn 4555  {cpr 4557  ‘cfv 6485  TopOnctopon 22893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-top 22877  df-topon 22894

This theorem is referenced by:  restsn2  23154  rrxtopn0  46736