Theorem vnex 4225

Description: The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)

Assertion

Ref	Expression
vnex	|- -. E. x x = _V

Proof of Theorem vnex

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nalset 4224	. 2 |- -. E. x A. y y e. x
2		vex 2806	. . . . . 6 |- y e. _V
3	2	tbt 247	. . . . 5 |- ( y e. x <-> ( y e. x <-> y e. _V ) )
4	3	albii 1519	. . . 4 |- ( A. y y e. x <-> A. y ( y e. x <-> y e. _V ) )
5		dfcleq 2225	. . . 4 |- ( x = _V <-> A. y ( y e. x <-> y e. _V ) )
6	4, 5	bitr4i 187	. . 3 |- ( A. y y e. x <-> x = _V )
7	6	exbii 1654	. 2 |- ( E. x A. y y e. x <-> E. x x = _V )
8	1, 7	mtbi 677	1 |- -. E. x x = _V

Syntax hints:   -. wn 3   <-> wb 105   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803

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-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2805

This theorem is referenced by:  vprc  4226