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Theorem topgele 11978
Description: The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
Assertion
Ref Expression
topgele  |-  ( J  e.  (TopOn `  X
)  ->  ( { (/)
,  X }  C_  J  /\  J  C_  ~P X ) )

Proof of Theorem topgele
StepHypRef Expression
1 topontop 11963 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 0opn 11955 . . . 4  |-  ( J  e.  Top  ->  (/)  e.  J
)
31, 2syl 14 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (/)  e.  J
)
4 toponmax 11974 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
5 0ex 3995 . . . 4  |-  (/)  e.  _V
6 prssg 3624 . . . 4  |-  ( (
(/)  e.  _V  /\  X  e.  J )  ->  (
( (/)  e.  J  /\  X  e.  J )  <->  {
(/) ,  X }  C_  J ) )
75, 4, 6sylancr 408 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( ( (/) 
e.  J  /\  X  e.  J )  <->  { (/) ,  X }  C_  J ) )
83, 4, 7mpbi2and 895 . 2  |-  ( J  e.  (TopOn `  X
)  ->  { (/) ,  X }  C_  J )
9 toponuni 11964 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
10 eqimss2 3102 . . . 4  |-  ( X  =  U. J  ->  U. J  C_  X )
119, 10syl 14 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  U. J  C_  X )
12 sspwuni 3843 . . 3  |-  ( J 
C_  ~P X  <->  U. J  C_  X )
1311, 12sylibr 133 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  C_  ~P X )
148, 13jca 302 1  |-  ( J  e.  (TopOn `  X
)  ->  ( { (/)
,  X }  C_  J  /\  J  C_  ~P X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1299    e. wcel 1448   _Vcvv 2641    C_ wss 3021   (/)c0 3310   ~Pcpw 3457   {cpr 3475   U.cuni 3683   ` cfv 5059   Topctop 11946  TopOnctopon 11959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-top 11947  df-topon 11960
This theorem is referenced by: (None)
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