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Theorem topgele 16954
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 16946 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 0opn 16932 . . . 4  |-  ( J  e.  Top  ->  (/)  e.  J
)
31, 2syl 16 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (/)  e.  J
)
4 toponmax 16948 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
5 0ex 4299 . . . 4  |-  (/)  e.  _V
6 prssg 3913 . . . 4  |-  ( (
(/)  e.  _V  /\  X  e.  J )  ->  (
( (/)  e.  J  /\  X  e.  J )  <->  {
(/) ,  X }  C_  J ) )
75, 4, 6sylancr 645 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( ( (/) 
e.  J  /\  X  e.  J )  <->  { (/) ,  X }  C_  J ) )
83, 4, 7mpbi2and 888 . 2  |-  ( J  e.  (TopOn `  X
)  ->  { (/) ,  X }  C_  J )
9 toponuni 16947 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
10 eqimss2 3361 . . . 4  |-  ( X  =  U. J  ->  U. J  C_  X )
119, 10syl 16 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  U. J  C_  X )
12 sspwuni 4136 . . 3  |-  ( J 
C_  ~P X  <->  U. J  C_  X )
1311, 12sylibr 204 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  C_  ~P X )
148, 13jca 519 1  |-  ( J  e.  (TopOn `  X
)  ->  ( { (/)
,  X }  C_  J  /\  J  C_  ~P X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {cpr 3775   U.cuni 3975   ` cfv 5413   Topctop 16913  TopOnctopon 16914
This theorem is referenced by:  topsn  16955  txindis  17619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-top 16918  df-topon 16921
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