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Theorem vprc 3879
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. (Contributed by NM, 23-Aug-1993.)
Assertion
Ref Expression
vprc ¬ V V

Proof of Theorem vprc
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nalset 3878 . . 3 ¬ xy y x
2 vex 2554 . . . . . . 7 y V
32tbt 236 . . . . . 6 (y x ↔ (y xy V))
43albii 1356 . . . . 5 (y y xy(y xy V))
5 dfcleq 2031 . . . . 5 (x = V ↔ y(y xy V))
64, 5bitr4i 176 . . . 4 (y y xx = V)
76exbii 1493 . . 3 (xy y xx x = V)
81, 7mtbi 594 . 2 ¬ x x = V
9 isset 2555 . 2 (V V ↔ x x = V)
108, 9mtbir 595 1 ¬ V V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  nvel  3880  vnex  3881  intexr  3895  intnexr  3896  snnex  4147  ruALT  4229  iprc  4543
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