Theorem nvelv 1483

Description: The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity.

Assertion

Ref	Expression
nvelv	|- -. V e. V

Proof of Theorem nvelv

Step	Hyp	Ref	Expression
1		nalset 1482	. . 3 |- -. E.xA.y y e. x
2		visset 1350	. . . . . . 7 |- y e. V
3	2	tbt 541	. . . . . 6 |- (y e. x <-> (y e. x <-> y e. V))
4	3	bial 695	. . . . 5 |- (A.y y e. x <-> A.y(y e. x <-> y e. V))
5		dfcleq 1098	. . . . 5 |- (x = V <-> A.y(y e. x <-> y e. V))
6	4, 5	bitr4 154	. . . 4 |- (A.y y e. x <-> x = V)
7	6	biex 733	. . 3 |- (E.xA.y y e. x <-> E.x x = V)
8	1, 7	mtbi 166	. 2 |- -. E.x x = V
9		isset 1351	. 2 |- (V e. V <-> E.x x = V)
10	8, 9	mtbir 167	1 |- -. V e. V

Syntax hints:  -. wn 1   <-> wb 127  A.wal 672  E.wex 678   e. wel 803   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  intex 1986

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-13 804  ax-14 805  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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