MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en1top Structured version   Visualization version   GIF version

Theorem en1top 22991
Description: {∅} is the only topology with one element. (Contributed by FL, 18-Aug-2008.)
Assertion
Ref Expression
en1top (𝐽 ∈ Top → (𝐽 ≈ 1o𝐽 = {∅}))

Proof of Theorem en1top
StepHypRef Expression
1 0opn 22910 . . 3 (𝐽 ∈ Top → ∅ ∈ 𝐽)
2 en1eqsn 9308 . . . 4 ((∅ ∈ 𝐽𝐽 ≈ 1o) → 𝐽 = {∅})
32ex 412 . . 3 (∅ ∈ 𝐽 → (𝐽 ≈ 1o𝐽 = {∅}))
41, 3syl 17 . 2 (𝐽 ∈ Top → (𝐽 ≈ 1o𝐽 = {∅}))
5 id 22 . . 3 (𝐽 = {∅} → 𝐽 = {∅})
6 0ex 5307 . . . 4 ∅ ∈ V
76ensn1 9061 . . 3 {∅} ≈ 1o
85, 7eqbrtrdi 5182 . 2 (𝐽 = {∅} → 𝐽 ≈ 1o)
94, 8impbid1 225 1 (𝐽 ∈ Top → (𝐽 ≈ 1o𝐽 = {∅}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  c0 4333  {csn 4626   class class class wbr 5143  1oc1o 8499  cen 8982  Topctop 22899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-en 8986  df-top 22900
This theorem is referenced by:  hmph0  23803
  Copyright terms: Public domain W3C validator