Theorem 0top 23005

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 868	. . 3 ⊢ (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅}))
2		0opn 22925	. . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
3		n0i 4345	. . . . . 6 ⊢ (∅ ∈ 𝐽 → ¬ 𝐽 = ∅)
4	2, 3	syl 17	. . . . 5 ⊢ (𝐽 ∈ Top → ¬ 𝐽 = ∅)
5	4	pm2.21d 121	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 = ∅ → 𝐽 = {∅}))
6		idd 24	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} → 𝐽 = {∅}))
7	5, 6	jaod 859	. . 3 ⊢ (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅}))
8	1, 7	impbid2 226	. 2 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})))
9		uni0b 4937	. . 3 ⊢ (∪ 𝐽 = ∅ ↔ 𝐽 ⊆ {∅})
10		sssn 4830	. . 3 ⊢ (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))
11	9, 10	bitr2i 276	. 2 ⊢ ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ ∪ 𝐽 = ∅)
12	8, 11	bitr2di 288	1 ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1536   ∈ wcel 2105   ⊆ wss 3962  ∅c0 4338  {csn 4630  ∪ cuni 4911  Topctop 22914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705  ax-sep 5301

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-in 3969  df-ss 3979  df-nul 4339  df-pw 4606  df-sn 4631  df-uni 4912  df-top 22915

This theorem is referenced by:  locfinref  33801