Theorem nvelv 2718

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 2717	. . 3 |- -. E.xA.y y e. x
2		visset 1816	. . . . . . 7 |- y e. V
3	2	tbt 722	. . . . . 6 |- (y e. x <-> (y e. x <-> y e. V))
4	3	albii 1001	. . . . 5 |- (A.y y e. x <-> A.y(y e. x <-> y e. V))
5		dfcleq 1473	. . . . 5 |- (x = V <-> A.y(y e. x <-> y e. V))
6	4, 5	bitr4 176	. . . 4 |- (A.y y e. x <-> x = V)
7	6	exbii 1053	. . 3 |- (E.xA.y y e. x <-> E.x x = V)
8	1, 7	mtbi 191	. 2 |- -. E.x x = V
9		isset 1817	. 2 |- (V e. V <-> E.x x = V)
10	8, 9	mtbir 192	1 |- -. V e. V

Syntax hints:  -. wn 2   <-> wb 146  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  Vcvv 1814

This theorem is referenced by:  nvel 2719  vnex 2720  intex 2734  intnex 2735  issetid 3286  inelv 3368  inpc 10476

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-8 966  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462  ax-sep 2708

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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