Theorem singcon 14968

Description: A singleton is a connected part of any topology.

Hypothesis

Ref	Expression
singcon.1	|- X = U.J

Assertion

Ref	Expression
singcon	|- ((J e. Top /\ A e. X) -> (subSp` <.{A}, J>.) e. Con)

Proof of Theorem singcon

Step	Hyp	Ref	Expression
1		singcon.1	. . . . . 6 |- X = U.J
2	1	subtopsin2 14907	. . . . 5 |- ((J e. Top /\ A e. X) -> (subSp` <.{A}, J>.) = {(/), {A}})
3		stoig2 10252	. . . . . . . 8 |- ((J e. Top /\ {A} C_ U.J) -> U.(subSp` <.{A}, J>.) = {A})
4	3	eqcomd 1889	. . . . . . 7 |- ((J e. Top /\ {A} C_ U.J) -> {A} = U.(subSp` <.{A}, J>.))
5		snssg 3124	. . . . . . . . 9 |- (A e. X -> (A e. U.J <-> {A} C_ U.J))
6	1	eleq2i 1961	. . . . . . . . 9 |- (A e. X <-> A e. U.J)
7	5, 6	syl5bb 591	. . . . . . . 8 |- (A e. X -> (A e. X <-> {A} C_ U.J))
8	7	ibi 652	. . . . . . 7 |- (A e. X -> {A} C_ U.J)
9	4, 8	sylan2 500	. . . . . 6 |- ((J e. Top /\ A e. X) -> {A} = U.(subSp` <.{A}, J>.))
10		preq2 3099	. . . . . 6 |- ({A} = U.(subSp` <.{A}, J>.) -> {(/), {A}} = {(/), U.(subSp` <.{A}, J>.)})
11	9, 10	syl 12	. . . . 5 |- ((J e. Top /\ A e. X) -> {(/), {A}} = {(/), U.(subSp` <.{A}, J>.)})
12	2, 11	eqtrd 1925	. . . 4 |- ((J e. Top /\ A e. X) -> (subSp` <.{A}, J>.) = {(/), U.(subSp` <.{A}, J>.)})
13	2	fveq2d 4685	. . . . 5 |- ((J e. Top /\ A e. X) -> (Clsd` (subSp` <.{A}, J>.)) = (Clsd` {(/), {A}}))
14		snex 3492	. . . . . . 7 |- {A} e. _V
15	14	clindistop 14962	. . . . . 6 |- (Clsd` {(/), {A}}) = {(/), {A}}
16	15	a1i 8	. . . . 5 |- ((J e. Top /\ A e. X) -> (Clsd` {(/), {A}}) = {(/), {A}})
17	3, 8	sylan2 500	. . . . . . 7 |- ((J e. Top /\ A e. X) -> U.(subSp` <.{A}, J>.) = {A})
18	17	eqcomd 1889	. . . . . 6 |- ((J e. Top /\ A e. X) -> {A} = U.(subSp` <.{A}, J>.))
19	18, 10	syl 12	. . . . 5 |- ((J e. Top /\ A e. X) -> {(/), {A}} = {(/), U.(subSp` <.{A}, J>.)})
20	13, 16, 19	3eqtrd 1929	. . . 4 |- ((J e. Top /\ A e. X) -> (Clsd` (subSp` <.{A}, J>.)) = {(/), U.(subSp` <.{A}, J>.)})
21	12, 20	ineq12d 2797	. . 3 |- ((J e. Top /\ A e. X) -> ((subSp` <.{A}, J>.) i^i (Clsd` (subSp` <.{A}, J>.))) = ({(/), U.(subSp` <.{A}, J>.)} i^i {(/), U.(subSp` <.{A}, J>.)}))
22		inidm 2803	. . 3 |- ({(/), U.(subSp` <.{A}, J>.)} i^i {(/), U.(subSp` <.{A}, J>.)}) = {(/), U.(subSp` <.{A}, J>.)}
23	21, 22	syl6eq 1944	. 2 |- ((J e. Top /\ A e. X) -> ((subSp` <.{A}, J>.) i^i (Clsd` (subSp` <.{A}, J>.))) = {(/), U.(subSp` <.{A}, J>.)})
24		stoig3 10253	. . . 4 |- ((J e. Top /\ {A} C_ U.J) -> (subSp` <.{A}, J>.) e. Top)
25	24, 8	sylan2 500	. . 3 |- ((J e. Top /\ A e. X) -> (subSp` <.{A}, J>.) e. Top)
26		iscon 10339	. . 3 |- ((subSp` <.{A}, J>.) e. Top -> ((subSp` <.{A}, J>.) e. Con <-> ((subSp` <.{A}, J>.) i^i (Clsd` (subSp` <.{A}, J>.))) = {(/), U.(subSp` <.{A}, J>.)}))
27	25, 26	syl 12	. 2 |- ((J e. Top /\ A e. X) -> ((subSp` <.{A}, J>.) e. Con <-> ((subSp` <.{A}, J>.) i^i (Clsd` (subSp` <.{A}, J>.))) = {(/), U.(subSp` <.{A}, J>.)}))
28	23, 27	mpbird 213	1 |- ((J e. Top /\ A e. X) -> (subSp` <.{A}, J>.) e. Con)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  {cpr 3045  <.cop 3046  U.cuni 3177  ` cfv 3998  Topctop 8857  Clsdccld 8936  subSpcsubsp 10242  Conccon 10337

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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  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-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  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-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-top 8861  df-topsp 8862  df-cld 8939  df-subsp 10243  df-con 10338

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