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Theorem vprc 4719
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 4718 . . 3 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 3175 . . . . . . 7 𝑦 ∈ V
32tbt 357 . . . . . 6 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1736 . . . . 5 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2603 . . . . 5 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 265 . . . 4 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1763 . . 3 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 310 . 2 ¬ ∃𝑥 𝑥 = V
9 isset 3179 . 2 (V ∈ V ↔ ∃𝑥 𝑥 = V)
108, 9mtbir 311 1 ¬ V ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wal 1472   = wceq 1474  wex 1694  wcel 1976  Vcvv 3172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-v 3174
This theorem is referenced by:  nvel  4720  vnex  4721  intex  4742  intnex  4743  snnex  6839  iprc  6970  elfi2  8180  fi0  8186  ruALT  8368  cardmin2  8684  00lsp  18748  bj-xnex  32041  fveqvfvv  39650  ndmaovcl  39730  opabn1stprc  40126
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