Theorem 0top 8905

Description: The singleton of the empty set is the only topology possible for an empty underlying set.

Assertion

Ref	Expression
0top	|- (J e. Top -> (U.J = (/) <-> J = {(/)}))

Proof of Theorem 0top

Step	Hyp	Ref	Expression
1		olc 290	. . 3 |- (J = {(/)} -> (J = (/) \/ J = {(/)}))
2		0opn 8870	. . . . . 6 |- (J e. Top -> (/) e. J)
3		n0i 2880	. . . . . 6 |- ((/) e. J -> -. J = (/))
4	2, 3	syl 12	. . . . 5 |- (J e. Top -> -. J = (/))
5	4	pm2.21d 94	. . . 4 |- (J e. Top -> (J = (/) -> J = {(/)}))
6		idd 75	. . . 4 |- (J e. Top -> (J = {(/)} -> J = {(/)}))
7	5, 6	jaod 469	. . 3 |- (J e. Top -> ((J = (/) \/ J = {(/)}) -> J = {(/)}))
8	1, 7	impbid2 576	. 2 |- (J e. Top -> (J = {(/)} <-> (J = (/) \/ J = {(/)})))
9		uni0b 3203	. . 3 |- (U.J = (/) <-> J C_ {(/)})
10		sssn 3142	. . 3 |- (J C_ {(/)} <-> (J = (/) \/ J = {(/)}))
11	9, 10	bitr2i 191	. 2 |- ((J = (/) \/ J = {(/)}) <-> U.J = (/))
12	8, 11	syl6rbb 596	1 |- (J e. Top -> (U.J = (/) <-> J = {(/)}))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   = wceq 1298   e. wcel 1300   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  Topctop 8857

This theorem is referenced by:  top2ind 14897

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-uni 3178  df-top 8861

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