Theorem indistop 7541

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 2749	. . 3 |- {(/), A} e. V
2		istopg 7489	. . 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 2445	. . . 4 |- (x (_ {(/), A} <-> ((x = (/) \/ x = {(/)}) \/ (x = {A} \/ x = {(/), A})))
5		unieq 2478	. . . . . . 7 |- (x = (/) -> U.x = U.(/))
6		uni0 2493	. . . . . . . 8 |- U.(/) = (/)
7		0ex 2679	. . . . . . . . 9 |- (/) e. V
8	7	pri1 2420	. . . . . . . 8 |- (/) e. {(/), A}
9	6, 8	eqeltr 1520	. . . . . . 7 |- U.(/) e. {(/), A}
10	5, 9	syl6eqel 1532	. . . . . 6 |- (x = (/) -> U.x e. {(/), A})
11		unieq 2478	. . . . . . 7 |- (x = {(/)} -> U.x = U.{(/)})
12	8	a1i 8	. . . . . . . 8 |- (x = {(/)} -> (/) e. {(/), A})
13	7	unisn 2485	. . . . . . . 8 |- U.{(/)} = (/)
14	12, 13	syl5eqel 1528	. . . . . . 7 |- (x = {(/)} -> U.{(/)} e. {(/), A})
15	11, 14	eqeltrd 1524	. . . . . 6 |- (x = {(/)} -> U.x e. {(/), A})
16	10, 15	jaoi 341	. . . . 5 |- ((x = (/) \/ x = {(/)}) -> U.x e. {(/), A})
17		unieq 2478	. . . . . . 7 |- (x = {A} -> U.x = U.{A})
18		indistop.1	. . . . . . . . . 10 |- A e. V
19	18	pri2 2421	. . . . . . . . 9 |- A e. {(/), A}
20	19	a1i 8	. . . . . . . 8 |- (x = {A} -> A e. {(/), A})
21	18	unisn 2485	. . . . . . . 8 |- U.{A} = A
22	20, 21	syl5eqel 1528	. . . . . . 7 |- (x = {A} -> U.{A} e. {(/), A})
23	17, 22	eqeltrd 1524	. . . . . 6 |- (x = {A} -> U.x e. {(/), A})
24		unieq 2478	. . . . . . 7 |- (x = {(/), A} -> U.x = U.{(/), A})
25		uncom 2147	. . . . . . . . . 10 |- (A u. (/)) = ((/) u. A)
26		un0 2268	. . . . . . . . . . 11 |- (A u. (/)) = A
27	26, 19	eqeltr 1520	. . . . . . . . . 10 |- (A u. (/)) e. {(/), A}
28	25, 27	eqeltrr 1521	. . . . . . . . 9 |- ((/) u. A) e. {(/), A}
29	28	a1i 8	. . . . . . . 8 |- (x = {(/), A} -> ((/) u. A) e. {(/), A})
30	7, 18	unipr 2483	. . . . . . . 8 |- U.{(/), A} = ((/) u. A)
31	29, 30	syl5eqel 1528	. . . . . . 7 |- (x = {(/), A} -> U.{(/), A} e. {(/), A})
32	24, 31	eqeltrd 1524	. . . . . 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 955	. 2 |- A.x(x (_ {(/), A} -> U.x e. {(/), A})
37		pm3.26 319	. . . . . . . . . . . . 13 |- ((y = (/) /\ x = (/)) -> y = (/))
38	37	ineq2d 2188	. . . . . . . . . . . 12 |- ((y = (/) /\ x = (/)) -> (x i^i y) = (x i^i (/)))
39		in0 2269	. . . . . . . . . . . 12 |- (x i^i (/)) = (/)
40	38, 39	syl6eq 1499	. . . . . . . . . . 11 |- ((y = (/) /\ x = (/)) -> (x i^i y) = (/))
41	40, 8	syl6eqel 1532	. . . . . . . . . 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 2187	. . . . . . . . . . . 12 |- ((y = A /\ x = (/)) -> (x i^i y) = ((/) i^i y))
45		incom 2179	. . . . . . . . . . . . 13 |- ((/) i^i y) = (y i^i (/))
46		in0 2269	. . . . . . . . . . . . 13 |- (y i^i (/)) = (/)
47	45, 46	eqtr 1471	. . . . . . . . . . . 12 |- ((/) i^i y) = (/)
48	44, 47	syl6eq 1499	. . . . . . . . . . 11 |- ((y = A /\ x = (/)) -> (x i^i y) = (/))
49	48, 8	syl6eqel 1532	. . . . . . . . . 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 2183	. . . . . . . . . . . . 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 2269	. . . . . . . . . . . 12 |- (A i^i (/)) = (/)
56	54, 55	syl6eq 1499	. . . . . . . . . . 11 |- ((y = (/) /\ x = A) -> (x i^i y) = (/))
57	56, 8	syl6eqel 1532	. . . . . . . . . 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 2183	. . . . . . . . . . . . 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 2193	. . . . . . . . . . . 12 |- (A i^i A) = A
62	60, 61	syl6eq 1499	. . . . . . . . . . 11 |- ((y = A /\ x = A) -> (x i^i y) = A)
63	62, 19	syl6eqel 1532	. . . . . . . . . 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 1788	. . . . . . 7 |- y e. V
69	68	elpr 2395	. . . . . 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 1268	. . . 4 |- ((x = (/) \/ x = A) -> A.y(y e. {(/), A} -> (x i^i y) e. {(/), A}))
72		visset 1788	. . . . 5 |- x e. V
73	72	elpr 2395	. . . 4 |- (x e. {(/), A} <-> (x = (/) \/ x = A))
74		df-ral 1625	. . . 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 1674	. 2 |- A.x e. {(/), A}A.y e. {(/), A} (x i^i y) e. {(/), A}
77	3, 36, 76	mpbir2an 727	1 |- {(/), A} e. Top

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 950   = wceq 1099   e. wcel 1105  A.wral 1621  Vcvv 1786   u. cun 2016   i^i cin 2017   (_ wss 2018  (/)c0 2251  {csn 2380  {cpr 2381  U.cuni 2471  Topctop 7481

This theorem is referenced by:  indistps 7546

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-uni 2472  df-top 7485

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