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Theorem topgele 16672
Description: The topologies over the same set have a greatest element (the discrete topology) and a 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 16664 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 0opn 16650 . . . 4  |-  ( J  e.  Top  ->  (/)  e.  J
)
31, 2syl 15 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (/)  e.  J
)
4 toponmax 16666 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
5 0ex 4150 . . . 4  |-  (/)  e.  _V
6 prssg 3770 . . . 4  |-  ( (
(/)  e.  _V  /\  X  e.  J )  ->  (
( (/)  e.  J  /\  X  e.  J )  <->  {
(/) ,  X }  C_  J ) )
75, 4, 6sylancr 644 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( ( (/) 
e.  J  /\  X  e.  J )  <->  { (/) ,  X }  C_  J ) )
83, 4, 7mpbi2and 887 . 2  |-  ( J  e.  (TopOn `  X
)  ->  { (/) ,  X }  C_  J )
9 toponuni 16665 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
10 eqimss2 3231 . . . 4  |-  ( X  =  U. J  ->  U. J  C_  X )
119, 10syl 15 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  U. J  C_  X )
12 sspwuni 3987 . . 3  |-  ( J 
C_  ~P X  <->  U. J  C_  X )
1311, 12sylibr 203 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  C_  ~P X )
148, 13jca 518 1  |-  ( J  e.  (TopOn `  X
)  ->  ( { (/)
,  X }  C_  J  /\  J  C_  ~P X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   {cpr 3641   U.cuni 3827   ` cfv 5255   Topctop 16631  TopOnctopon 16632
This theorem is referenced by:  topsn  16673  txindis  17328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-topon 16639
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