Theorem 0top 23109

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

Step	Hyp	Ref	Expression
1		olc 881	. . 3 ⊢ (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅}))
2		0opn 23030	. . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
3		n0i 4301	. . . . . 6 ⊢ (∅ ∈ 𝐽 → ¬ 𝐽 = ∅)
4	2, 3	syl 18	. . . . 5 ⊢ (𝐽 ∈ Top → ¬ 𝐽 = ∅)
5	4	pm2.21d 122	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 = ∅ → 𝐽 = {∅}))
6		idd 25	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} → 𝐽 = {∅}))
7	5, 6	jaod 872	. . 3 ⊢ (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅}))
8	1, 7	impbid2 229	. 2 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})))
9		uni0b 4903	. . 3 ⊢ (∪ 𝐽 = ∅ ↔ 𝐽 ⊆ {∅})
10		sssn 4796	. . 3 ⊢ (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))
11	9, 10	bitr2i 279	. 2 ⊢ ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ ∪ 𝐽 = ∅)
12	8, 11	bitr2di 291	1 ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  ∅c0 4294  {csn 4594  ∪ cuni 4876  Topctop 23019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920  df-ss 3930  df-nul 4295  df-pw 4569  df-sn 4595  df-uni 4877  df-top 23020

This theorem is referenced by:  locfinref  34176