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Theorem 0top 22977
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 866 . . 3 (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅}))
2 0opn 22897 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ 𝐽)
3 n0i 4336 . . . . . 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 857 . . 3 (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅}))
81, 7impbid2 225 . 2 (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})))
9 uni0b 4941 . . 3 ( 𝐽 = ∅ ↔ 𝐽 ⊆ {∅})
10 sssn 4835 . . 3 (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))
119, 10bitr2i 275 . 2 ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ 𝐽 = ∅)
128, 11bitr2di 287 1 (𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wo 845   = wceq 1534  wcel 2099  wss 3947  c0 4325  {csn 4633   cuni 4913  Topctop 22886
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-11 2147  ax-ext 2697  ax-sep 5304
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4326  df-pw 4609  df-sn 4634  df-uni 4914  df-top 22887
This theorem is referenced by:  locfinref  33656
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