Theorem sn0top 7540

Description: The singleton of the empty set is a topology. (Contributed by Stefan Allan, 3-Mar-2006.)

Assertion

Ref	Expression
sn0top	|- {(/)} e. Top

Proof of Theorem sn0top

Step	Hyp	Ref	Expression
1		p0ex 2738	. . 3 |- {(/)} e. V
2		istopg 7489	. . 3 |- ({(/)} e. V -> ({(/)} e. Top <-> (A.x(x (_ {(/)} -> U.x e. {(/)}) /\ A.x e. {(/)}A.y e. {(/)} (x i^i y) e. {(/)})))
3	1, 2	ax-mp 7	. 2 |- ({(/)} e. Top <-> (A.x(x (_ {(/)} -> U.x e. {(/)}) /\ A.x e. {(/)}A.y e. {(/)} (x i^i y) e. {(/)}))
4		sssn 2443	. . . 4 |- (x (_ {(/)} <-> (x = (/) \/ x = {(/)}))
5		unieq 2478	. . . . . 6 |- (x = (/) -> U.x = U.(/))
6		uni0 2493	. . . . . . 7 |- U.(/) = (/)
7		0ex 2679	. . . . . . . 8 |- (/) e. V
8	7	elsnc2 2408	. . . . . . 7 |- (U.(/) e. {(/)} <-> U.(/) = (/))
9	6, 8	mpbir 190	. . . . . 6 |- U.(/) e. {(/)}
10	5, 9	syl6eqel 1532	. . . . 5 |- (x = (/) -> U.x e. {(/)})
11		unieq 2478	. . . . . 6 |- (x = {(/)} -> U.x = U.{(/)})
12	7	unisn 2485	. . . . . . . 8 |- U.{(/)} = (/)
13		eqtrt 1468	. . . . . . . 8 |- ((U.x = U.{(/)} /\ U.{(/)} = (/)) -> U.x = (/))
14	12, 13	mpan2 693	. . . . . . 7 |- (U.x = U.{(/)} -> U.x = (/))
15		visset 1788	. . . . . . . . 9 |- x e. V
16	15	uniex 2834	. . . . . . . 8 |- U.x e. V
17	16	elsnc 2402	. . . . . . 7 |- (U.x e. {(/)} <-> U.x = (/))
18	14, 17	sylibr 200	. . . . . 6 |- (U.x = U.{(/)} -> U.x e. {(/)})
19	11, 18	syl 10	. . . . 5 |- (x = {(/)} -> U.x e. {(/)})
20	10, 19	jaoi 341	. . . 4 |- ((x = (/) \/ x = {(/)}) -> U.x e. {(/)})
21	4, 20	sylbi 199	. . 3 |- (x (_ {(/)} -> U.x e. {(/)})
22	21	ax-gen 955	. 2 |- A.x(x (_ {(/)} -> U.x e. {(/)})
23		elsn 2392	. . . . 5 |- (y e. {(/)} <-> y = (/))
24		ineq2 2182	. . . . . . 7 |- (y = (/) -> (x i^i y) = (x i^i (/)))
25		in0 2269	. . . . . . . . 9 |- (x i^i (/)) = (/)
26	25	eqeq2i 1461	. . . . . . . 8 |- ((x i^i y) = (x i^i (/)) <-> (x i^i y) = (/))
27	26	biimp 151	. . . . . . 7 |- ((x i^i y) = (x i^i (/)) -> (x i^i y) = (/))
28	24, 27	syl 10	. . . . . 6 |- (y = (/) -> (x i^i y) = (/))
29	15	inex1 2684	. . . . . . . 8 |- (x i^i y) e. V
30	29	elsnc 2402	. . . . . . 7 |- ((x i^i y) e. {(/)} <-> (x i^i y) = (/))
31	30	biimpr 152	. . . . . 6 |- ((x i^i y) = (/) -> (x i^i y) e. {(/)})
32	28, 31	syl 10	. . . . 5 |- (y = (/) -> (x i^i y) e. {(/)})
33	23, 32	sylbi 199	. . . 4 |- (y e. {(/)} -> (x i^i y) e. {(/)})
34	33	adantl 388	. . 3 |- ((x e. {(/)} /\ y e. {(/)}) -> (x i^i y) e. {(/)})
35	34	rgen2a 1675	. 2 |- A.x e. {(/)}A.y e. {(/)} (x i^i y) e. {(/)}
36	3, 22, 35	mpbir2an 727	1 |- {(/)} 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   i^i cin 2017   (_ wss 2018  (/)c0 2251  {csn 2380  U.cuni 2471  Topctop 7481

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-un 2830

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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