Theorem indistop 7590

Description: The indiscrete topology on a set A. Part of Example 2 in [Munkres] p. 77. (Contributed by FL, 16-Jul-2006.)

Hypothesis

Ref	Expression
indistop.1	|- A e. V

Assertion

Ref	Expression
indistop	|- {(/), A} e. Top

Proof of Theorem indistop

Step	Hyp	Ref	Expression
1		prex 2771	. . 3 |- {(/), A} e. V
2		istopg 7538	. . 3 |- ({(/), A} e. V -> ({(/), A} e. Top <-> (A.x(x (_ {(/), A} -> U.x e. {(/), A}) /\ A.x e. {(/), A}A.y e. {(/), A} (x i^i y) e. {(/), A})))
3	1, 2	ax-mp 7	. 2 |- ({(/), A} e. Top <-> (A.x(x (_ {(/), A} -> U.x e. {(/), A}) /\ A.x e. {(/), A}A.y e. {(/), A} (x i^i y) e. {(/), A}))
4		sspr 2466	. . . 4 |- (x (_ {(/), A} <-> ((x = (/) \/ x = {(/)}) \/ (x = {A} \/ x = {(/), A})))
5		unieq 2500	. . . . . . 7 |- (x = (/) -> U.x = U.(/))
6		uni0 2515	. . . . . . . 8 |- U.(/) = (/)
7		0ex 2701	. . . . . . . . 9 |- (/) e. V
8	7	pri1 2441	. . . . . . . 8 |- (/) e. {(/), A}
9	6, 8	eqeltr 1536	. . . . . . 7 |- U.(/) e. {(/), A}
10	5, 9	syl6eqel 1548	. . . . . 6 |- (x = (/) -> U.x e. {(/), A})
11		unieq 2500	. . . . . . 7 |- (x = {(/)} -> U.x = U.{(/)})
12	8	a1i 8	. . . . . . . 8 |- (x = {(/)} -> (/) e. {(/), A})
13	7	unisn 2507	. . . . . . . 8 |- U.{(/)} = (/)
14	12, 13	syl5eqel 1544	. . . . . . 7 |- (x = {(/)} -> U.{(/)} e. {(/), A})
15	11, 14	eqeltrd 1540	. . . . . 6 |- (x = {(/)} -> U.x e. {(/), A})
16	10, 15	jaoi 341	. . . . 5 |- ((x = (/) \/ x = {(/)}) -> U.x e. {(/), A})
17		unieq 2500	. . . . . . 7 |- (x = {A} -> U.x = U.{A})
18		indistop.1	. . . . . . . . . 10 |- A e. V
19	18	pri2 2442	. . . . . . . . 9 |- A e. {(/), A}
20	19	a1i 8	. . . . . . . 8 |- (x = {A} -> A e. {(/), A})
21	18	unisn 2507	. . . . . . . 8 |- U.{A} = A
22	20, 21	syl5eqel 1544	. . . . . . 7 |- (x = {A} -> U.{A} e. {(/), A})
23	17, 22	eqeltrd 1540	. . . . . 6 |- (x = {A} -> U.x e. {(/), A})
24		unieq 2500	. . . . . . 7 |- (x = {(/), A} -> U.x = U.{(/), A})
25		uncom 2166	. . . . . . . . . 10 |- (A u. (/)) = ((/) u. A)
26		un0 2287	. . . . . . . . . . 11 |- (A u. (/)) = A
27	26, 19	eqeltr 1536	. . . . . . . . . 10 |- (A u. (/)) e. {(/), A}
28	25, 27	eqeltrr 1537	. . . . . . . . 9 |- ((/) u. A) e. {(/), A}
29	28	a1i 8	. . . . . . . 8 |- (x = {(/), A} -> ((/) u. A) e. {(/), A})
30	7, 18	unipr 2505	. . . . . . . 8 |- U.{(/), A} = ((/) u. A)
31	29, 30	syl5eqel 1544	. . . . . . 7 |- (x = {(/), A} -> U.{(/), A} e. {(/), A})
32	24, 31	eqeltrd 1540	. . . . . 6 |- (x = {(/), A} -> U.x e. {(/), A})
33	23, 32	jaoi 341	. . . . 5 |- ((x = {A} \/ x = {(/), A}) -> U.x e. {(/), A})
34	16, 33	jaoi 341	. . . 4 |- (((x = (/) \/ x = {(/)}) \/ (x = {A} \/ x = {(/), A})) -> U.x e. {(/), A})
35	4, 34	sylbi 199	. . 3 |- (x (_ {(/), A} -> U.x e. {(/), A})
36	35	ax-gen 960	. 2 |- A.x(x (_ {(/), A} -> U.x e. {(/), A})
37		pm3.26 319	. . . . . . . . . . . . 13 |- ((y = (/) /\ x = (/)) -> y = (/))
38	37	ineq2d 2207	. . . . . . . . . . . 12 |- ((y = (/) /\ x = (/)) -> (x i^i y) = (x i^i (/)))
39		in0 2288	. . . . . . . . . . . 12 |- (x i^i (/)) = (/)
40	38, 39	syl6eq 1515	. . . . . . . . . . 11 |- ((y = (/) /\ x = (/)) -> (x i^i y) = (/))
41	40, 8	syl6eqel 1548	. . . . . . . . . 10 |- ((y = (/) /\ x = (/)) -> (x i^i y) e. {(/), A})
42	41	ex 373	. . . . . . . . 9 |- (y = (/) -> (x = (/) -> (x i^i y) e. {(/), A}))
43		pm3.27 323	. . . . . . . . . . . . 13 |- ((y = A /\ x = (/)) -> x = (/))
44	43	ineq1d 2206	. . . . . . . . . . . 12 |- ((y = A /\ x = (/)) -> (x i^i y) = ((/) i^i y))
45		incom 2198	. . . . . . . . . . . . 13 |- ((/) i^i y) = (y i^i (/))
46		in0 2288	. . . . . . . . . . . . 13 |- (y i^i (/)) = (/)
47	45, 46	eqtr 1487	. . . . . . . . . . . 12 |- ((/) i^i y) = (/)
48	44, 47	syl6eq 1515	. . . . . . . . . . 11 |- ((y = A /\ x = (/)) -> (x i^i y) = (/))
49	48, 8	syl6eqel 1548	. . . . . . . . . 10 |- ((y = A /\ x = (/)) -> (x i^i y) e. {(/), A})
50	49	ex 373	. . . . . . . . 9 |- (y = A -> (x = (/) -> (x i^i y) e. {(/), A}))
51	42, 50	jaoi 341	. . . . . . . 8 |- ((y = (/) \/ y = A) -> (x = (/) -> (x i^i y) e. {(/), A}))
52	51	com12 11	. . . . . . 7 |- (x = (/) -> ((y = (/) \/ y = A) -> (x i^i y) e. {(/), A}))
53		ineq12 2202	. . . . . . . . . . . . 13 |- ((x = A /\ y = (/)) -> (x i^i y) = (A i^i (/)))
54	53	ancoms 436	. . . . . . . . . . . 12 |- ((y = (/) /\ x = A) -> (x i^i y) = (A i^i (/)))
55		in0 2288	. . . . . . . . . . . 12 |- (A i^i (/)) = (/)
56	54, 55	syl6eq 1515	. . . . . . . . . . 11 |- ((y = (/) /\ x = A) -> (x i^i y) = (/))
57	56, 8	syl6eqel 1548	. . . . . . . . . 10 |- ((y = (/) /\ x = A) -> (x i^i y) e. {(/), A})
58	57	ex 373	. . . . . . . . 9 |- (y = (/) -> (x = A -> (x i^i y) e. {(/), A}))
59		ineq12 2202	. . . . . . . . . . . . 13 |- ((x = A /\ y = A) -> (x i^i y) = (A i^i A))
60	59	ancoms 436	. . . . . . . . . . . 12 |- ((y = A /\ x = A) -> (x i^i y) = (A i^i A))
61		inidm 2212	. . . . . . . . . . . 12 |- (A i^i A) = A
62	60, 61	syl6eq 1515	. . . . . . . . . . 11 |- ((y = A /\ x = A) -> (x i^i y) = A)
63	62, 19	syl6eqel 1548	. . . . . . . . . 10 |- ((y = A /\ x = A) -> (x i^i y) e. {(/), A})
64	63	ex 373	. . . . . . . . 9 |- (y = A -> (x = A -> (x i^i y) e. {(/), A}))
65	58, 64	jaoi 341	. . . . . . . 8 |- ((y = (/) \/ y = A) -> (x = A -> (x i^i y) e. {(/), A}))
66	65	com12 11	. . . . . . 7 |- (x = A -> ((y = (/) \/ y = A) -> (x i^i y) e. {(/), A}))
67	52, 66	jaoi 341	. . . . . 6 |- ((x = (/) \/ x = A) -> ((y = (/) \/ y = A) -> (x i^i y) e. {(/), A}))
68		visset 1804	. . . . . . 7 |- y e. V
69	68	elpr 2414	. . . . . 6 |- (y e. {(/), A} <-> (y = (/) \/ y = A))
70	67, 69	syl5ib 206	. . . . 5 |- ((x = (/) \/ x = A) -> (y e. {(/), A} -> (x i^i y) e. {(/), A}))
71	70	19.21aiv 1281	. . . 4 |- ((x = (/) \/ x = A) -> A.y(y e. {(/), A} -> (x i^i y) e. {(/), A}))
72		visset 1804	. . . . 5 |- x e. V
73	72	elpr 2414	. . . 4 |- (x e. {(/), A} <-> (x = (/) \/ x = A))
74		df-ral 1641	. . . 4 |- (A.y e. {(/), A} (x i^i y) e. {(/), A} <-> A.y(y e. {(/), A} -> (x i^i y) e. {(/), A}))
75	71, 73, 74	3imtr4 219	. . 3 |- (x e. {(/), A} -> A.y e. {(/), A} (x i^i y) e. {(/), A})
76	75	rgen 1690	. 2 |- A.x e. {(/), A}A.y e. {(/), A} (x i^i y) e. {(/), A}
77	3, 36, 76	mpbir2an 728	1 |- {(/), A} e. Top

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802   u. cun 2035   i^i cin 2036   (_ wss 2037  (/)c0 2270  {csn 2399  {cpr 2400  U.cuni 2493  Topctop 7530

This theorem is referenced by:  indistps 7595  mapudiscn 10399

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-uni 2494  df-top 7534

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