Theorem 0top 20787

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 e. Top -> ( U. J = (/) <-> J = { (/) } ) )

Proof of Theorem 0top

Step	Hyp	Ref	Expression
1		olc 399	. . 3 |- ( J = { (/) } -> ( J = (/) \/ J = { (/) } ) )
2		0opn 20709	. . . . . 6 |- ( J e. Top -> (/) e. J )
3		n0i 3920	. . . . . 6 |- ( (/) e. J -> -. J = (/) )
4	2, 3	syl 17	. . . . 5 |- ( J e. Top -> -. J = (/) )
5	4	pm2.21d 118	. . . 4 |- ( J e. Top -> ( J = (/) -> J = { (/) } ) )
6		idd 24	. . . 4 |- ( J e. Top -> ( J = { (/) } -> J = { (/) } ) )
7	5, 6	jaod 395	. . 3 |- ( J e. Top -> ( ( J = (/) \/ J = { (/) } ) -> J = { (/) } ) )
8	1, 7	impbid2 216	. 2 |- ( J e. Top -> ( J = { (/) } <-> ( J = (/) \/ J = { (/) } ) ) )
9		uni0b 4463	. . 3 |- ( U. J = (/) <-> J C_ { (/) } )
10		sssn 4358	. . 3 |- ( J C_ { (/) } <-> ( J = (/) \/ J = { (/) } ) )
11	9, 10	bitr2i 265	. 2 |- ( ( J = (/) \/ J = { (/) } ) <-> U. J = (/) )
12	8, 11	syl6rbb 277	1 |- ( J e. Top -> ( U. J = (/) <-> J = { (/) } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   = wceq 1483   e. wcel 1990   C_ wss 3574   (/)c0 3915   {csn 4177   U.cuni 4436   Topctop 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-uni 4437  df-top 20699

This theorem is referenced by:  locfinref  29908