Theorem singempcon 14965

Description: The singleton of the empty set is a connected topology.

Assertion

Ref	Expression
singempcon	|- {(/)} e. Con

Proof of Theorem singempcon

Step	Hyp	Ref	Expression
1		ss0 2902	. . . . . . . 8 |- (x C_ (/) -> x = (/))
2	1	adantr 425	. . . . . . 7 |- ((x C_ (/) /\ ((/) \ x) e. {(/)}) -> x = (/))
3		eqimss 2665	. . . . . . . 8 |- (x = (/) -> x C_ (/))
4		difeq2 2719	. . . . . . . . 9 |- (x = (/) -> ((/) \ x) = ((/) \ (/)))
5		difid 2942	. . . . . . . . . 10 |- ((/) \ (/)) = (/)
6		0ex 3446	. . . . . . . . . . 11 |- (/) e. _V
7	6	snid 3069	. . . . . . . . . 10 |- (/) e. {(/)}
8	5, 7	eqeltri 1967	. . . . . . . . 9 |- ((/) \ (/)) e. {(/)}
9	4, 8	syl6eqel 1979	. . . . . . . 8 |- (x = (/) -> ((/) \ x) e. {(/)})
10	3, 9	jca 310	. . . . . . 7 |- (x = (/) -> (x C_ (/) /\ ((/) \ x) e. {(/)}))
11	2, 10	impbii 174	. . . . . 6 |- ((x C_ (/) /\ ((/) \ x) e. {(/)}) <-> x = (/))
12	11	abbii 2006	. . . . 5 |- {x | (x C_ (/) /\ ((/) \ x) e. {(/)})} = {x | x = (/)}
13		sn0top 8917	. . . . . 6 |- {(/)} e. Top
14	6	unisn 3193	. . . . . . . 8 |- U.{(/)} = (/)
15	14	eqcomi 1888	. . . . . . 7 |- (/) = U.{(/)}
16	15	cldval 8942	. . . . . 6 |- ({(/)} e. Top -> (Clsd` {(/)}) = {x | (x C_ (/) /\ ((/) \ x) e. {(/)})})
17	13, 16	ax-mp 7	. . . . 5 |- (Clsd` {(/)}) = {x | (x C_ (/) /\ ((/) \ x) e. {(/)})}
18		df-sn 3049	. . . . 5 |- {(/)} = {x | x = (/)}
19	12, 17, 18	3eqtr4i 1921	. . . 4 |- (Clsd` {(/)}) = {(/)}
20	19	ineq2i 2793	. . 3 |- ({(/)} i^i (Clsd` {(/)})) = ({(/)} i^i {(/)})
21		inidm 2803	. . 3 |- ({(/)} i^i {(/)}) = {(/)}
22		dfsn2 3057	. . . 4 |- {(/)} = {(/), (/)}
23		preq2 3099	. . . . 5 |- ((/) = U.{(/)} -> {(/), (/)} = {(/), U.{(/)}})
24	15, 23	ax-mp 7	. . . 4 |- {(/), (/)} = {(/), U.{(/)}}
25	22, 24	eqtri 1908	. . 3 |- {(/)} = {(/), U.{(/)}}
26	20, 21, 25	3eqtri 1912	. 2 |- ({(/)} i^i (Clsd` {(/)})) = {(/), U.{(/)}}
27		iscon 10339	. . 3 |- ({(/)} e. Top -> ({(/)} e. Con <-> ({(/)} i^i (Clsd` {(/)})) = {(/), U.{(/)}}))
28	13, 27	ax-mp 7	. 2 |- ({(/)} e. Con <-> ({(/)} i^i (Clsd` {(/)})) = {(/), U.{(/)}})
29	26, 28	mpbir 207	1 |- {(/)} e. Con

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871   \ cdif 2590   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  {cpr 3045  U.cuni 3177  ` cfv 3998  Topctop 8857  Clsdccld 8936  Conccon 10337

This theorem is referenced by:  empcon 14966

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-13 1311  ax-14 1312  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  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-top 8861  df-cld 8939  df-con 10338

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