Theorem topnex 14809

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4546. (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 4546	. . . 4 |- { y | E. x y = ~P x } e/ _V
2	1	neli 2499	. . 3 |- -. { y | E. x y = ~P x } e. _V
3		vex 2805	. . . . . . . 8 |- x e. _V
4		distop 14808	. . . . . . . 8 |- ( x e. _V -> ~P x e. Top )
5	3, 4	ax-mp 5	. . . . . . 7 |- ~P x e. Top
6		eleq1 2294	. . . . . . 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 1646	. . . . 5 |- ( E. x y = ~P x -> y e. Top )
9	8	abssi 3302	. . . 4 |- { y | E. x y = ~P x } C_ Top
10		ssexg 4228	. . . 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 668	. 2 |- -. Top e. _V
13	12	nelir 2500	1 |- Top e/ _V

Syntax hints:   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   e/ wnel 2497   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   Topctop 14720

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-nel 2498  df-ral 2515  df-rex 2516  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-uni 3894  df-iun 3972  df-top 14721

This theorem is referenced by: (None)