Theorem 0top 21593

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 864	. . 3 ⊢ (𝐽 = {∅} → (𝐽 = ∅ ∨ 𝐽 = {∅}))
2		0opn 21514	. . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
3		n0i 4301	. . . . . 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 855	. . 3 ⊢ (𝐽 ∈ Top → ((𝐽 = ∅ ∨ 𝐽 = {∅}) → 𝐽 = {∅}))
8	1, 7	impbid2 228	. 2 ⊢ (𝐽 ∈ Top → (𝐽 = {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅})))
9		uni0b 4866	. . 3 ⊢ (∪ 𝐽 = ∅ ↔ 𝐽 ⊆ {∅})
10		sssn 4761	. . 3 ⊢ (𝐽 ⊆ {∅} ↔ (𝐽 = ∅ ∨ 𝐽 = {∅}))
11	9, 10	bitr2i 278	. 2 ⊢ ((𝐽 = ∅ ∨ 𝐽 = {∅}) ↔ ∪ 𝐽 = ∅)
12	8, 11	syl6rbb 290	1 ⊢ (𝐽 ∈ Top → (∪ 𝐽 = ∅ ↔ 𝐽 = {∅}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ⊆ wss 3938  ∅c0 4293  {csn 4569  ∪ cuni 4840  Topctop 21503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-in 3945  df-ss 3954  df-nul 4294  df-pw 4543  df-sn 4570  df-uni 4841  df-top 21504

This theorem is referenced by:  locfinref  31107