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Theorem topgele 12210
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 12195 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
2 0opn 12187 . . . 4  |-  ( J  e.  Top  ->  (/)  e.  J
)
31, 2syl 14 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  (/)  e.  J
)
4 toponmax 12206 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
5 0ex 4055 . . . 4  |-  (/)  e.  _V
6 prssg 3677 . . . 4  |-  ( (
(/)  e.  _V  /\  X  e.  J )  ->  (
( (/)  e.  J  /\  X  e.  J )  <->  {
(/) ,  X }  C_  J ) )
75, 4, 6sylancr 410 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  ( ( (/) 
e.  J  /\  X  e.  J )  <->  { (/) ,  X }  C_  J ) )
83, 4, 7mpbi2and 927 . 2  |-  ( J  e.  (TopOn `  X
)  ->  { (/) ,  X }  C_  J )
9 toponuni 12196 . . . 4  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
10 eqimss2 3152 . . . 4  |-  ( X  =  U. J  ->  U. J  C_  X )
119, 10syl 14 . . 3  |-  ( J  e.  (TopOn `  X
)  ->  U. J  C_  X )
12 sspwuni 3897 . . 3  |-  ( J 
C_  ~P X  <->  U. J  C_  X )
1311, 12sylibr 133 . 2  |-  ( J  e.  (TopOn `  X
)  ->  J  C_  ~P X )
148, 13jca 304 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 1331    e. wcel 1480   _Vcvv 2686    C_ wss 3071   (/)c0 3363   ~Pcpw 3510   {cpr 3528   U.cuni 3736   ` cfv 5123   Topctop 12178  TopOnctopon 12191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-top 12179  df-topon 12192
This theorem is referenced by: (None)
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