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Theorem sn0top 7589
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
StepHypRef Expression
1 p0ex 2760 . . 3 |- {(/)} e. V
2 istopg 7538 . . 3 |- ({(/)} e. V -> ({(/)} e. Top <-> (A.x(x (_ {(/)} -> U.x e. {(/)}) /\ A.x e. {(/)}A.y e. {(/)} (x i^i y) e. {(/)})))
31, 2ax-mp 7 . 2 |- ({(/)} e. Top <-> (A.x(x (_ {(/)} -> U.x e. {(/)}) /\ A.x e. {(/)}A.y e. {(/)} (x i^i y) e. {(/)}))
4 sssn 2464 . . . 4 |- (x (_ {(/)} <-> (x = (/) \/ x = {(/)}))
5 unieq 2500 . . . . . 6 |- (x = (/) -> U.x = U.(/))
6 uni0 2515 . . . . . . 7 |- U.(/) = (/)
7 0ex 2701 . . . . . . . 8 |- (/) e. V
87elsnc2 2427 . . . . . . 7 |- (U.(/) e. {(/)} <-> U.(/) = (/))
96, 8mpbir 190 . . . . . 6 |- U.(/) e. {(/)}
105, 9syl6eqel 1548 . . . . 5 |- (x = (/) -> U.x e. {(/)})
11 unieq 2500 . . . . . 6 |- (x = {(/)} -> U.x = U.{(/)})
127unisn 2507 . . . . . . . 8 |- U.{(/)} = (/)
13 eqtrt 1484 . . . . . . . 8 |- ((U.x = U.{(/)} /\ U.{(/)} = (/)) -> U.x = (/))
1412, 13mpan2 694 . . . . . . 7 |- (U.x = U.{(/)} -> U.x = (/))
15 visset 1804 . . . . . . . . 9 |- x e. V
1615uniex 2861 . . . . . . . 8 |- U.x e. V
1716elsnc 2421 . . . . . . 7 |- (U.x e. {(/)} <-> U.x = (/))
1814, 17sylibr 200 . . . . . 6 |- (U.x = U.{(/)} -> U.x e. {(/)})
1911, 18syl 10 . . . . 5 |- (x = {(/)} -> U.x e. {(/)})
2010, 19jaoi 341 . . . 4 |- ((x = (/) \/ x = {(/)}) -> U.x e. {(/)})
214, 20sylbi 199 . . 3 |- (x (_ {(/)} -> U.x e. {(/)})
2221ax-gen 960 . 2 |- A.x(x (_ {(/)} -> U.x e. {(/)})
23 elsn 2411 . . . . 5 |- (y e. {(/)} <-> y = (/))
24 ineq2 2201 . . . . . . 7 |- (y = (/) -> (x i^i y) = (x i^i (/)))
25 in0 2288 . . . . . . . . 9 |- (x i^i (/)) = (/)
2625eqeq2i 1477 . . . . . . . 8 |- ((x i^i y) = (x i^i (/)) <-> (x i^i y) = (/))
2726biimp 151 . . . . . . 7 |- ((x i^i y) = (x i^i (/)) -> (x i^i y) = (/))
2824, 27syl 10 . . . . . 6 |- (y = (/) -> (x i^i y) = (/))
2915inex1 2706 . . . . . . . 8 |- (x i^i y) e. V
3029elsnc 2421 . . . . . . 7 |- ((x i^i y) e. {(/)} <-> (x i^i y) = (/))
3130biimpr 152 . . . . . 6 |- ((x i^i y) = (/) -> (x i^i y) e. {(/)})
3228, 31syl 10 . . . . 5 |- (y = (/) -> (x i^i y) e. {(/)})
3323, 32sylbi 199 . . . 4 |- (y e. {(/)} -> (x i^i y) e. {(/)})
3433adantl 388 . . 3 |- ((x e. {(/)} /\ y e. {(/)}) -> (x i^i y) e. {(/)})
3534rgen2a 1691 . 2 |- A.x e. {(/)}A.y e. {(/)} (x i^i y) e. {(/)}
363, 22, 35mpbir2an 728 1 |- {(/)} e. Top
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802   i^i cin 2036   (_ wss 2037  (/)c0 2270  {csn 2399  U.cuni 2493  Topctop 7530
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-un 2857
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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