Theorem topsn 22826

Description: The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4896). (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 22825	. . 3 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → ({∅, {𝐴}} ⊆ 𝐽 ∧ 𝐽 ⊆ 𝒫 {𝐴}))
2	1	simprd 495	. 2 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 ⊆ 𝒫 {𝐴})
3		pwsn 4896	. . 3 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
4	1	simpld 494	. . 3 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → {∅, {𝐴}} ⊆ 𝐽)
5	3, 4	eqsstrid 4026	. 2 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝒫 {𝐴} ⊆ 𝐽)
6	2, 5	eqssd 3995	1 ⊢ (𝐽 ∈ (TopOn‘{𝐴}) → 𝐽 = 𝒫 {𝐴})

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   ⊆ wss 3945  ∅c0 4318  𝒫 cpw 4598  {csn 4624  {cpr 4626  ‘cfv 6542  TopOnctopon 22805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-top 22789  df-topon 22806

This theorem is referenced by:  restsn2  23068  rrxtopn0  45675