Theorem clindistop 14962

Description: The closed sets of an indiscrete topology.

Hypothesis

Ref	Expression
clindistop.1	|- A e. B

Assertion

Ref	Expression
clindistop	|- (Clsd` {(/), A}) = {(/), A}

Proof of Theorem clindistop

Step	Hyp	Ref	Expression
1		indistop 8918	. 2 |- {(/), A} e. Top
2		0ex 3446	. . . . . 6 |- (/) e. _V
3		clindistop.1	. . . . . . 7 |- A e. B
4	3	elisseti 2301	. . . . . 6 |- A e. _V
5	2, 4	unipr 3191	. . . . 5 |- U.{(/), A} = ((/) u. A)
6		uncom 2744	. . . . 5 |- (A u. (/)) = ((/) u. A)
7		un0 2896	. . . . 5 |- (A u. (/)) = A
8	5, 6, 7	3eqtr2ri 1916	. . . 4 |- A = U.{(/), A}
9	8	cldval 8942	. . 3 |- ({(/), A} e. Top -> (Clsd` {(/), A}) = {x | (x C_ A /\ (A \ x) e. {(/), A})})
10		difexg 3458	. . . . . . . . . 10 |- (A e. B -> (A \ x) e. _V)
11	3, 10	ax-mp 7	. . . . . . . . 9 |- (A \ x) e. _V
12	11	elpr 3061	. . . . . . . 8 |- ((A \ x) e. {(/), A} <-> ((A \ x) = (/) \/ (A \ x) = A))
13		ssdif0 2934	. . . . . . . . . 10 |- (A C_ x <-> (A \ x) = (/))
14		eqss 2631	. . . . . . . . . . . . 13 |- (x = A <-> (x C_ A /\ A C_ x))
15	14	biimpri 169	. . . . . . . . . . . 12 |- ((x C_ A /\ A C_ x) -> x = A)
16	15	olcd 295	. . . . . . . . . . 11 |- ((x C_ A /\ A C_ x) -> (x = (/) \/ x = A))
17	16	expcom 403	. . . . . . . . . 10 |- (A C_ x -> (x C_ A -> (x = (/) \/ x = A)))
18	13, 17	sylbir 218	. . . . . . . . 9 |- ((A \ x) = (/) -> (x C_ A -> (x = (/) \/ x = A)))
19		disj3 2918	. . . . . . . . . . 11 |- ((A i^i x) = (/) <-> A = (A \ x))
20		incom 2787	. . . . . . . . . . . 12 |- (A i^i x) = (x i^i A)
21		reldisj 2916	. . . . . . . . . . . . . . . 16 |- (x C_ A -> ((x i^i A) = (/) <-> x C_ (A \ A)))
22		difid 2942	. . . . . . . . . . . . . . . . . . 19 |- (A \ A) = (/)
23	22	sseq2i 2642	. . . . . . . . . . . . . . . . . 18 |- (x C_ (A \ A) <-> x C_ (/))
24		ss0 2902	. . . . . . . . . . . . . . . . . 18 |- (x C_ (/) -> x = (/))
25	23, 24	sylbi 216	. . . . . . . . . . . . . . . . 17 |- (x C_ (A \ A) -> x = (/))
26	25	orcd 294	. . . . . . . . . . . . . . . 16 |- (x C_ (A \ A) -> (x = (/) \/ x = A))
27	21, 26	syl6bi 231	. . . . . . . . . . . . . . 15 |- (x C_ A -> ((x i^i A) = (/) -> (x = (/) \/ x = A)))
28		eqtr 1904	. . . . . . . . . . . . . . 15 |- (((x i^i A) = (A i^i x) /\ (A i^i x) = (/)) -> (x i^i A) = (/))
29	27, 28	syl5com 63	. . . . . . . . . . . . . 14 |- (((x i^i A) = (A i^i x) /\ (A i^i x) = (/)) -> (x C_ A -> (x = (/) \/ x = A)))
30	29	ex 402	. . . . . . . . . . . . 13 |- ((x i^i A) = (A i^i x) -> ((A i^i x) = (/) -> (x C_ A -> (x = (/) \/ x = A))))
31	30	eqcoms 1887	. . . . . . . . . . . 12 |- ((A i^i x) = (x i^i A) -> ((A i^i x) = (/) -> (x C_ A -> (x = (/) \/ x = A))))
32	20, 31	ax-mp 7	. . . . . . . . . . 11 |- ((A i^i x) = (/) -> (x C_ A -> (x = (/) \/ x = A)))
33	19, 32	sylbir 218	. . . . . . . . . 10 |- (A = (A \ x) -> (x C_ A -> (x = (/) \/ x = A)))
34	33	eqcoms 1887	. . . . . . . . 9 |- ((A \ x) = A -> (x C_ A -> (x = (/) \/ x = A)))
35	18, 34	jaoi 368	. . . . . . . 8 |- (((A \ x) = (/) \/ (A \ x) = A) -> (x C_ A -> (x = (/) \/ x = A)))
36	12, 35	sylbi 216	. . . . . . 7 |- ((A \ x) e. {(/), A} -> (x C_ A -> (x = (/) \/ x = A)))
37	36	impcom 378	. . . . . 6 |- ((x C_ A /\ (A \ x) e. {(/), A}) -> (x = (/) \/ x = A))
38		0ss 2900	. . . . . . . . 9 |- (/) C_ A
39		dif0 2943	. . . . . . . . . 10 |- (A \ (/)) = A
40	4	prid2 3107	. . . . . . . . . 10 |- A e. {(/), A}
41	39, 40	eqeltri 1967	. . . . . . . . 9 |- (A \ (/)) e. {(/), A}
42	38, 41	pm3.2i 307	. . . . . . . 8 |- ((/) C_ A /\ (A \ (/)) e. {(/), A})
43		sseq1 2637	. . . . . . . . 9 |- (x = (/) -> (x C_ A <-> (/) C_ A))
44		difeq2 2719	. . . . . . . . . 10 |- (x = (/) -> (A \ x) = (A \ (/)))
45	44	eleq1d 1963	. . . . . . . . 9 |- (x = (/) -> ((A \ x) e. {(/), A} <-> (A \ (/)) e. {(/), A}))
46	43, 45	anbi12d 690	. . . . . . . 8 |- (x = (/) -> ((x C_ A /\ (A \ x) e. {(/), A}) <-> ((/) C_ A /\ (A \ (/)) e. {(/), A})))
47	42, 46	mpbiri 211	. . . . . . 7 |- (x = (/) -> (x C_ A /\ (A \ x) e. {(/), A}))
48		ssid 2634	. . . . . . . . 9 |- A C_ A
49	2	prid1 3106	. . . . . . . . . 10 |- (/) e. {(/), A}
50	22, 49	eqeltri 1967	. . . . . . . . 9 |- (A \ A) e. {(/), A}
51	48, 50	pm3.2i 307	. . . . . . . 8 |- (A C_ A /\ (A \ A) e. {(/), A})
52		sseq1 2637	. . . . . . . . 9 |- (x = A -> (x C_ A <-> A C_ A))
53		difeq2 2719	. . . . . . . . . 10 |- (x = A -> (A \ x) = (A \ A))
54	53	eleq1d 1963	. . . . . . . . 9 |- (x = A -> ((A \ x) e. {(/), A} <-> (A \ A) e. {(/), A}))
55	52, 54	anbi12d 690	. . . . . . . 8 |- (x = A -> ((x C_ A /\ (A \ x) e. {(/), A}) <-> (A C_ A /\ (A \ A) e. {(/), A})))
56	51, 55	mpbiri 211	. . . . . . 7 |- (x = A -> (x C_ A /\ (A \ x) e. {(/), A}))
57	47, 56	jaoi 368	. . . . . 6 |- ((x = (/) \/ x = A) -> (x C_ A /\ (A \ x) e. {(/), A}))
58	37, 57	impbii 174	. . . . 5 |- ((x C_ A /\ (A \ x) e. {(/), A}) <-> (x = (/) \/ x = A))
59	58	abbii 2006	. . . 4 |- {x | (x C_ A /\ (A \ x) e. {(/), A})} = {x | (x = (/) \/ x = A)}
60		dfpr2 3059	. . . 4 |- {(/), A} = {x | (x = (/) \/ x = A)}
61	59, 60	eqtr4i 1911	. . 3 |- {x | (x C_ A /\ (A \ x) e. {(/), A})} = {(/), A}
62	9, 61	syl6eq 1944	. 2 |- ({(/), A} e. Top -> (Clsd` {(/), A}) = {(/), A})
63	1, 62	ax-mp 7	1 |- (Clsd` {(/), A}) = {(/), A}

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  {cpr 3045  U.cuni 3177  ` cfv 3998  Topctop 8857  Clsdccld 8936

This theorem is referenced by:  clindistop2 14963  intopcon 14964  singcon 14968

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

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