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Theorem topnex 12182
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4340. (Contributed by BJ, 2-May-2021.)
Assertion
Ref Expression
topnex  |-  Top  e/  _V

Proof of Theorem topnex
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwnex 4340 . . . 4  |-  { y  |  E. x  y  =  ~P x }  e/  _V
21neli 2382 . . 3  |-  -.  {
y  |  E. x  y  =  ~P x }  e.  _V
3 vex 2663 . . . . . . . 8  |-  x  e. 
_V
4 distop 12181 . . . . . . . 8  |-  ( x  e.  _V  ->  ~P x  e.  Top )
53, 4ax-mp 5 . . . . . . 7  |-  ~P x  e.  Top
6 eleq1 2180 . . . . . . 7  |-  ( y  =  ~P x  -> 
( y  e.  Top  <->  ~P x  e.  Top )
)
75, 6mpbiri 167 . . . . . 6  |-  ( y  =  ~P x  -> 
y  e.  Top )
87exlimiv 1562 . . . . 5  |-  ( E. x  y  =  ~P x  ->  y  e.  Top )
98abssi 3142 . . . 4  |-  { y  |  E. x  y  =  ~P x }  C_ 
Top
10 ssexg 4037 . . . 4  |-  ( ( { y  |  E. x  y  =  ~P x }  C_  Top  /\  Top  e.  _V )  ->  { y  |  E. x  y  =  ~P x }  e.  _V )
119, 10mpan 420 . . 3  |-  ( Top 
e.  _V  ->  { y  |  E. x  y  =  ~P x }  e.  _V )
122, 11mto 636 . 2  |-  -.  Top  e.  _V
1312nelir 2383 1  |-  Top  e/  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1316   E.wex 1453    e. wcel 1465   {cab 2103    e/ wnel 2380   _Vcvv 2660    C_ wss 3041   ~Pcpw 3480   Topctop 12091
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-nel 2381  df-ral 2398  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-uni 3707  df-iun 3785  df-top 12092
This theorem is referenced by: (None)
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