Theorem topnex 15110

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4590. (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 4590	. . . 4 |- { y | E. x y = ~P x } e/ _V
2	1	neli 2517	. . 3 |- -. { y | E. x y = ~P x } e. _V
3		vex 2824	. . . . . . . 8 |- x e. _V
4		distop 15109	. . . . . . . 8 |- ( x e. _V -> ~P x e. Top )
5	3, 4	ax-mp 5	. . . . . . 7 |- ~P x e. Top
6		eleq1 2301	. . . . . . 7 |- ( y = ~P x -> ( y e. Top <-> ~P x e. Top ) )
7	5, 6	mpbiri 168	. . . . . 6 |- ( y = ~P x -> y e. Top )
8	7	exlimiv 1651	. . . . 5 |- ( E. x y = ~P x -> y e. Top )
9	8	abssi 3323	. . . 4 |- { y | E. x y = ~P x } C_ Top
10		ssexg 4267	. . . 4 |- ( ( { y | E. x y = ~P x } C_ Top /\ Top e. _V ) -> { y | E. x y = ~P x } e. _V )
11	9, 10	mpan 428	. . 3 |- ( Top e. _V -> { y | E. x y = ~P x } e. _V )
12	2, 11	mto 672	. 2 |- -. Top e. _V
13	12	nelir 2518	1 |- Top e/ _V

Syntax hints:   = wceq 1402   E.wex 1545   e. wcel 2209   {cab 2224   e/ wnel 2515   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685   Topctop 15021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-nel 2516  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-uni 3931  df-iun 4009  df-top 15022

This theorem is referenced by: (None)