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Theorem 0top 20698
 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 399 . . 3 (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅}))
2 0opn 20634 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ 𝐽)
3 n0i 3896 . . . . . 6 (∅ ∈ 𝐽 → ¬ 𝐽 = ∅)
42, 3syl 17 . . . . 5 (𝐽 ∈ Top → ¬ 𝐽 = ∅)
54pm2.21d 118 . . . 4 (𝐽 ∈ Top → (𝐽 = ∅ → 𝐽 = {∅}))
6 idd 24 . . . 4 (𝐽 ∈ Top → (𝐽 = {∅} → 𝐽 = {∅}))
75, 6jaod 395 . . 3 (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅}))
81, 7impbid2 216 . 2 (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})))
9 uni0b 4429 . . 3 ( 𝐽 = ∅ ↔ 𝐽 ⊆ {∅})
10 sssn 4326 . . 3 (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))
119, 10bitr2i 265 . 2 ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ 𝐽 = ∅)
128, 11syl6rbb 277 1 (𝐽 ∈ Top → ( 𝐽 = ∅ ↔ 𝐽 = {∅}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   = wceq 1480   ∈ wcel 1987   ⊆ wss 3555  ∅c0 3891  {csn 4148  ∪ cuni 4402  Topctop 20617 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132  df-sn 4149  df-uni 4403  df-top 20621 This theorem is referenced by:  locfinref  29687
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