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Theorem topgele 17004
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 16996 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 0opn 16982 . . . 4  |-  ( J  e.  Top  ->  (/)  e.  J
)
31, 2syl 16 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (/)  e.  J
)
4 toponmax 16998 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
5 0ex 4342 . . . 4  |-  (/)  e.  _V
6 prssg 3955 . . . 4  |-  ( (
(/)  e.  _V  /\  X  e.  J )  ->  (
( (/)  e.  J  /\  X  e.  J )  <->  {
(/) ,  X }  C_  J ) )
75, 4, 6sylancr 646 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( ( (/) 
e.  J  /\  X  e.  J )  <->  { (/) ,  X }  C_  J ) )
83, 4, 7mpbi2and 889 . 2  |-  ( J  e.  (TopOn `  X
)  ->  { (/) ,  X }  C_  J )
9 toponuni 16997 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
10 eqimss2 3403 . . . 4  |-  ( X  =  U. J  ->  U. J  C_  X )
119, 10syl 16 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  U. J  C_  X )
12 sspwuni 4179 . . 3  |-  ( J 
C_  ~P X  <->  U. J  C_  X )
1311, 12sylibr 205 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  C_  ~P X )
148, 13jca 520 1  |-  ( J  e.  (TopOn `  X
)  ->  ( { (/)
,  X }  C_  J  /\  J  C_  ~P X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   {cpr 3817   U.cuni 4017   ` cfv 5457   Topctop 16963  TopOnctopon 16964
This theorem is referenced by:  topsn  17005  txindis  17671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-top 16968  df-topon 16971
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