Theorem topnex 20800

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 6968; an alternate proof uses indiscrete topologies (see indistop 20806) and the analogue of pwnex 6968 with pairs { (/) , x } instead of power sets ~P x (that analogue is also a consequence of abnex 6965). (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 6968	. . . 4 |- { y | E. x y = ~P x } e/ _V
2	1	neli 2899	. . 3 |- -. { y | E. x y = ~P x } e. _V
3		vex 3203	. . . . . . . 8 |- x e. _V
4		distop 20799	. . . . . . . 8 |- ( x e. _V -> ~P x e. Top )
5	3, 4	ax-mp 5	. . . . . . 7 |- ~P x e. Top
6		eleq1 2689	. . . . . . 7 |- ( y = ~P x -> ( y e. Top <-> ~P x e. Top ) )
7	5, 6	mpbiri 248	. . . . . 6 |- ( y = ~P x -> y e. Top )
8	7	exlimiv 1858	. . . . 5 |- ( E. x y = ~P x -> y e. Top )
9	8	abssi 3677	. . . 4 |- { y | E. x y = ~P x } C_ Top
10		ssexg 4804	. . . 4 |- ( ( { y | E. x y = ~P x } C_ Top /\ Top e. _V ) -> { y | E. x y = ~P x } e. _V )
11	9, 10	mpan 706	. . 3 |- ( Top e. _V -> { y | E. x y = ~P x } e. _V )
12	2, 11	mto 188	. 2 |- -. Top e. _V
13	12	nelir 2900	1 |- Top e/ _V

Syntax hints:   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   e/ wnel 2897   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   Topctop 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-iun 4522  df-top 20699

This theorem is referenced by: (None)