Theorem sn0top 8917

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 3495	. . 3 |- {(/)} e. _V
2		istopg 8865	. . 3 |- ({(/)} e. _V -> ({(/)} e. Top <-> (A.x(x C_ {(/)} -> 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 C_ {(/)} -> U.x e. {(/)}) /\ A.x e. {(/)}A.y e. {(/)} (x i^i y) e. {(/)}))
4		sssn 3142	. . . 4 |- (x C_ {(/)} <-> (x = (/) \/ x = {(/)}))
5		unieq 3185	. . . . . 6 |- (x = (/) -> U.x = U.(/))
6		uni0 3205	. . . . . . 7 |- U.(/) = (/)
7		0ex 3446	. . . . . . . 8 |- (/) e. _V
8	7	elsnc2 3071	. . . . . . 7 |- (U.(/) e. {(/)} <-> U.(/) = (/))
9	6, 8	mpbir 207	. . . . . 6 |- U.(/) e. {(/)}
10	5, 9	syl6eqel 1979	. . . . 5 |- (x = (/) -> U.x e. {(/)})
11		unieq 3185	. . . . . 6 |- (x = {(/)} -> U.x = U.{(/)})
12	7	unisn 3193	. . . . . . . 8 |- U.{(/)} = (/)
13		eqtr 1904	. . . . . . . 8 |- ((U.x = U.{(/)} /\ U.{(/)} = (/)) -> U.x = (/))
14	12, 13	mpan2 760	. . . . . . 7 |- (U.x = U.{(/)} -> U.x = (/))
15		visset 2295	. . . . . . . . 9 |- x e. _V
16	15	uniex 3794	. . . . . . . 8 |- U.x e. _V
17	16	elsnc 3065	. . . . . . 7 |- (U.x e. {(/)} <-> U.x = (/))
18	14, 17	sylibr 217	. . . . . 6 |- (U.x = U.{(/)} -> U.x e. {(/)})
19	11, 18	syl 12	. . . . 5 |- (x = {(/)} -> U.x e. {(/)})
20	10, 19	jaoi 368	. . . 4 |- ((x = (/) \/ x = {(/)}) -> U.x e. {(/)})
21	4, 20	sylbi 216	. . 3 |- (x C_ {(/)} -> U.x e. {(/)})
22	21	ax-gen 1305	. 2 |- A.x(x C_ {(/)} -> U.x e. {(/)})
23		elsn 3058	. . . . 5 |- (y e. {(/)} <-> y = (/))
24		ineq2 2790	. . . . . . 7 |- (y = (/) -> (x i^i y) = (x i^i (/)))
25		in0 2897	. . . . . . . . 9 |- (x i^i (/)) = (/)
26	25	eqeq2i 1894	. . . . . . . 8 |- ((x i^i y) = (x i^i (/)) <-> (x i^i y) = (/))
27	26	biimpi 168	. . . . . . 7 |- ((x i^i y) = (x i^i (/)) -> (x i^i y) = (/))
28	24, 27	syl 12	. . . . . 6 |- (y = (/) -> (x i^i y) = (/))
29	15	inex1 3452	. . . . . . . 8 |- (x i^i y) e. _V
30	29	elsnc 3065	. . . . . . 7 |- ((x i^i y) e. {(/)} <-> (x i^i y) = (/))
31	30	biimpri 169	. . . . . 6 |- ((x i^i y) = (/) -> (x i^i y) e. {(/)})
32	28, 31	syl 12	. . . . 5 |- (y = (/) -> (x i^i y) e. {(/)})
33	23, 32	sylbi 216	. . . 4 |- (y e. {(/)} -> (x i^i y) e. {(/)})
34	33	adantl 424	. . 3 |- ((x e. {(/)} /\ y e. {(/)}) -> (x i^i y) e. {(/)})
35	34	rgen2a 2160	. 2 |- A.x e. {(/)}A.y e. {(/)} (x i^i y) e. {(/)}
36	3, 22, 35	mpbir2an 800	1 |- {(/)} e. Top

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  Topctop 8857

This theorem is referenced by:  indistop 8918  issubspt 10247  stoig 10251  homindlem2 14899  subsp2 14902  subspemp2 14904  sbtpsines 14905  sinempcomp 14953  singempcon 14965  txcnoprab 15911

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-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  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-uni 3178  df-top 8861

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