Theorem topnex 14760

Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4540. (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 4540	. . . 4 |- { y | E. x y = ~P x } e/ _V
2	1	neli 2497	. . 3 |- -. { y | E. x y = ~P x } e. _V
3		vex 2802	. . . . . . . 8 |- x e. _V
4		distop 14759	. . . . . . . 8 |- ( x e. _V -> ~P x e. Top )
5	3, 4	ax-mp 5	. . . . . . 7 |- ~P x e. Top
6		eleq1 2292	. . . . . . 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 1644	. . . . 5 |- ( E. x y = ~P x -> y e. Top )
9	8	abssi 3299	. . . 4 |- { y | E. x y = ~P x } C_ Top
10		ssexg 4223	. . . 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 666	. 2 |- -. Top e. _V
13	12	nelir 2498	1 |- Top e/ _V

Syntax hints:   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   e/ wnel 2495   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   Topctop 14671

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-nel 2496  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-uni 3889  df-iun 3967  df-top 14672

This theorem is referenced by: (None)