Theorem topnex 14673

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4514. (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 4514	. . . 4 |- { y | E. x y = ~P x } e/ _V
2	1	neli 2475	. . 3 |- -. { y | E. x y = ~P x } e. _V
3		vex 2779	. . . . . . . 8 |- x e. _V
4		distop 14672	. . . . . . . 8 |- ( x e. _V -> ~P x e. Top )
5	3, 4	ax-mp 5	. . . . . . 7 |- ~P x e. Top
6		eleq1 2270	. . . . . . 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 1622	. . . . 5 |- ( E. x y = ~P x -> y e. Top )
9	8	abssi 3276	. . . 4 |- { y | E. x y = ~P x } C_ Top
10		ssexg 4199	. . . 4 |- ( ( { y | E. x y = ~P x } C_ Top /\ Top e. _V ) -> { y | E. x y = ~P x } e. _V )
11	9, 10	mpan 424	. . 3 |- ( Top e. _V -> { y | E. x y = ~P x } e. _V )
12	2, 11	mto 664	. 2 |- -. Top e. _V
13	12	nelir 2476	1 |- Top e/ _V

Syntax hints:   = wceq 1373   E.wex 1516   e. wcel 2178   {cab 2193   e/ wnel 2473   _Vcvv 2776   C_ wss 3174   ~Pcpw 3626   Topctop 14584

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-nel 2474  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-uni 3865  df-iun 3943  df-top 14585

This theorem is referenced by: (None)