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Theorem 0top 21116
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 895 . . 3 (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅}))
2 0opn 21037 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ 𝐽)
3 n0i 4120 . . . . . 6 (∅ ∈ 𝐽 → ¬ 𝐽 = ∅)
42, 3syl 17 . . . . 5 (𝐽 ∈ Top → ¬ 𝐽 = ∅)
54pm2.21d 119 . . . 4 (𝐽 ∈ Top → (𝐽 = ∅ → 𝐽 = {∅}))
6 idd 24 . . . 4 (𝐽 ∈ Top → (𝐽 = {∅} → 𝐽 = {∅}))
75, 6jaod 886 . . 3 (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅}))
81, 7impbid2 218 . 2 (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})))
9 uni0b 4655 . . 3 ( 𝐽 = ∅ ↔ 𝐽 ⊆ {∅})
10 sssn 4545 . . 3 (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))
119, 10bitr2i 268 . 2 ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ 𝐽 = ∅)
128, 11syl6rbb 280 1 (𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wo 874   = wceq 1653  wcel 2157  wss 3769  c0 4115  {csn 4368   cuni 4628  Topctop 21026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-v 3387  df-dif 3772  df-in 3776  df-ss 3783  df-nul 4116  df-pw 4351  df-sn 4369  df-uni 4629  df-top 21027
This theorem is referenced by:  locfinref  30424
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