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.

Step	Hyp	Ref	Expression
1		pwnex 4308	. . . 4 |- { y | E. x y = ~P x } e/ _V
2	1	neli 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 )
5	3, 4	ax-mp 7	. . . . . . 7 |- ~P x e. Top
6		eleq1 2162	. . . . . . 7 |- ( y = ~P x -> ( y e. Top <-> ~P x e. Top ) )
7	5, 6	mpbiri 167	. . . . . 6 |- ( y = ~P x -> y e. Top )
8	7	exlimiv 1545	. . . . 5 |- ( E. x y = ~P x -> y e. Top )
9	8	abssi 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 )
11	9, 10	mpan 418	. . 3 |- ( Top e. _V -> { y | E. x y = ~P x } e. _V )
12	2, 11	mto 629	. 2 |- -. Top e. _V
13	12	nelir 2365	1 |- Top e/ _V

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)