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Theorem 0top 22349
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 867 . . 3 (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅}))
2 0opn 22269 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ 𝐽)
3 n0i 4294 . . . . . 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 858 . . 3 (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅}))
81, 7impbid2 225 . 2 (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})))
9 uni0b 4895 . . 3 ( 𝐽 = ∅ ↔ 𝐽 ⊆ {∅})
10 sssn 4787 . . 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 205  wo 846   = wceq 1542  wcel 2107  wss 3911  c0 4283  {csn 4587   cuni 4866  Topctop 22258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-in 3918  df-ss 3928  df-nul 4284  df-pw 4563  df-sn 4588  df-uni 4867  df-top 22259
This theorem is referenced by:  locfinref  32479
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