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Theorem topnex 12037
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4308. (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 4308 . . . 4  |-  { y  |  E. x  y  =  ~P x }  e/  _V
21neli 2364 . . 3  |-  -.  {
y  |  E. x  y  =  ~P x }  e.  _V
3 vex 2644 . . . . . . . 8  |-  x  e. 
_V
4 distop 12036 . . . . . . . 8  |-  ( x  e.  _V  ->  ~P x  e.  Top )
53, 4ax-mp 7 . . . . . . 7  |-  ~P x  e.  Top
6 eleq1 2162 . . . . . . 7  |-  ( y  =  ~P x  -> 
( y  e.  Top  <->  ~P x  e.  Top )
)
75, 6mpbiri 167 . . . . . 6  |-  ( y  =  ~P x  -> 
y  e.  Top )
87exlimiv 1545 . . . . 5  |-  ( E. x  y  =  ~P x  ->  y  e.  Top )
98abssi 3119 . . . 4  |-  { y  |  E. x  y  =  ~P x }  C_ 
Top
10 ssexg 4007 . . . 4  |-  ( ( { y  |  E. x  y  =  ~P x }  C_  Top  /\  Top  e.  _V )  ->  { y  |  E. x  y  =  ~P x }  e.  _V )
119, 10mpan 418 . . 3  |-  ( Top 
e.  _V  ->  { y  |  E. x  y  =  ~P x }  e.  _V )
122, 11mto 629 . 2  |-  -.  Top  e.  _V
1312nelir 2365 1  |-  Top  e/  _V
Colors of variables: wff set class
Syntax hints:    = wceq 1299   E.wex 1436    e. wcel 1448   {cab 2086    e/ wnel 2362   _Vcvv 2641    C_ wss 3021   ~Pcpw 3457   Topctop 11946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-nel 2363  df-ral 2380  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-uni 3684  df-iun 3762  df-top 11947
This theorem is referenced by: (None)
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