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Theorem 0top 22990
Description: The singleton of the empty set is the only topology possible for an empty underlying set. (Contributed by NM, 9-Sep-2006.)
Assertion
Ref Expression
0top (𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))

Proof of Theorem 0top
StepHypRef Expression
1 olc 869 . . 3 (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅}))
2 0opn 22910 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ 𝐽)
3 n0i 4340 . . . . . 6 (∅ ∈ 𝐽 → ¬ 𝐽 = ∅)
42, 3syl 17 . . . . 5 (𝐽 ∈ Top → ¬ 𝐽 = ∅)
54pm2.21d 121 . . . 4 (𝐽 ∈ Top → (𝐽 = ∅ → 𝐽 = {∅}))
6 idd 24 . . . 4 (𝐽 ∈ Top → (𝐽 = {∅} → 𝐽 = {∅}))
75, 6jaod 860 . . 3 (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅}))
81, 7impbid2 226 . 2 (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})))
9 uni0b 4933 . . 3 ( 𝐽 = ∅ ↔ 𝐽 ⊆ {∅})
10 sssn 4826 . . 3 (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))
119, 10bitr2i 276 . 2 ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ 𝐽 = ∅)
128, 11bitr2di 288 1 (𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wo 848   = wceq 1540  wcel 2108  wss 3951  c0 4333  {csn 4626   cuni 4907  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-11 2157  ax-ext 2708  ax-sep 5296
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-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-uni 4908  df-top 22900
This theorem is referenced by:  locfinref  33840
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