Theorem topsinind 14967

Description: The only topology on a singleton is the indiscrete topology.

Hypothesis

Ref	Expression
topsinind.1	|- A e. B

Assertion

Ref	Expression
topsinind	|- (J e. Top -> (U.J = {A} <-> J = {(/), {A}}))

Proof of Theorem topsinind

Step	Hyp	Ref	Expression
1		topindis 14859	. 2 |- (J e. Top -> ({(/), U.J} C_ J /\ J C_ ~PU.J))
2		pweq 3036	. . . . . . . . 9 |- (U.J = {A} -> ~PU.J = ~P{A})
3	2	sseq2d 2645	. . . . . . . 8 |- (U.J = {A} -> (J C_ ~PU.J <-> J C_ ~P{A}))
4		preq2 3099	. . . . . . . . 9 |- (U.J = {A} -> {(/), U.J} = {(/), {A}})
5	4	sseq1d 2644	. . . . . . . 8 |- (U.J = {A} -> ({(/), U.J} C_ J <-> {(/), {A}} C_ J))
6	3, 5	anbi12d 690	. . . . . . 7 |- (U.J = {A} -> ((J C_ ~PU.J /\ {(/), U.J} C_ J) <-> (J C_ ~P{A} /\ {(/), {A}} C_ J)))
7		pwsn 3172	. . . . . . . 8 |- ~P{A} = {(/), {A}}
8		sseq2 2639	. . . . . . . . . 10 |- (~P{A} = {(/), {A}} -> (J C_ ~P{A} <-> J C_ {(/), {A}}))
9	8	anbi1d 679	. . . . . . . . 9 |- (~P{A} = {(/), {A}} -> ((J C_ ~P{A} /\ {(/), {A}} C_ J) <-> (J C_ {(/), {A}} /\ {(/), {A}} C_ J)))
10		eqss 2631	. . . . . . . . . 10 |- (J = {(/), {A}} <-> (J C_ {(/), {A}} /\ {(/), {A}} C_ J))
11	10	biimpri 169	. . . . . . . . 9 |- ((J C_ {(/), {A}} /\ {(/), {A}} C_ J) -> J = {(/), {A}})
12	9, 11	syl6bi 231	. . . . . . . 8 |- (~P{A} = {(/), {A}} -> ((J C_ ~P{A} /\ {(/), {A}} C_ J) -> J = {(/), {A}}))
13	7, 12	ax-mp 7	. . . . . . 7 |- ((J C_ ~P{A} /\ {(/), {A}} C_ J) -> J = {(/), {A}})
14	6, 13	syl6bi 231	. . . . . 6 |- (U.J = {A} -> ((J C_ ~PU.J /\ {(/), U.J} C_ J) -> J = {(/), {A}}))
15	14	exp3a 405	. . . . 5 |- (U.J = {A} -> (J C_ ~PU.J -> ({(/), U.J} C_ J -> J = {(/), {A}})))
16	15	com13 37	. . . 4 |- ({(/), U.J} C_ J -> (J C_ ~PU.J -> (U.J = {A} -> J = {(/), {A}})))
17	16	imp 377	. . 3 |- (({(/), U.J} C_ J /\ J C_ ~PU.J) -> (U.J = {A} -> J = {(/), {A}}))
18		unieq 3185	. . . 4 |- (J = {(/), {A}} -> U.J = U.{(/), {A}})
19		0ex 3446	. . . . . 6 |- (/) e. _V
20		snex 3492	. . . . . 6 |- {A} e. _V
21	19, 20	unipr 3191	. . . . 5 |- U.{(/), {A}} = ((/) u. {A})
22		uncom 2744	. . . . 5 |- ({A} u. (/)) = ((/) u. {A})
23		un0 2896	. . . . 5 |- ({A} u. (/)) = {A}
24	21, 22, 23	3eqtr2i 1915	. . . 4 |- U.{(/), {A}} = {A}
25	18, 24	syl6eq 1944	. . 3 |- (J = {(/), {A}} -> U.J = {A})
26	17, 25	impbid1 575	. 2 |- (({(/), U.J} C_ J /\ J C_ ~PU.J) -> (U.J = {A} <-> J = {(/), {A}}))
27	1, 26	syl 12	1 |- (J e. Top -> (U.J = {A} <-> J = {(/), {A}}))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   u. cun 2591   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  {csn 3044  {cpr 3045  U.cuni 3177  Topctop 8857

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  ax-reg 5695

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-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-xp 4000  df-rel 4001  df-top 8861  df-topsp 8862

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