Theorem 0top 21588

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 865	. . 3 ⊢ (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅}))
2		0opn 21509	. . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
3		n0i 4249	. . . . . 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 856	. . 3 ⊢ (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅}))
8	1, 7	impbid2 229	. 2 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})))
9		uni0b 4826	. . 3 ⊢ (∪ 𝐽 = ∅ ↔ 𝐽 ⊆ {∅})
10		sssn 4719	. . 3 ⊢ (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))
11	9, 10	bitr2i 279	. 2 ⊢ ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ ∪ 𝐽 = ∅)
12	8, 11	syl6rbb 291	1 ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∅c0 4243  {csn 4525  ∪ cuni 4800  Topctop 21498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-nul 4244  df-pw 4499  df-sn 4526  df-uni 4801  df-top 21499

This theorem is referenced by:  locfinref  31194