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Theorem vnex 4930
Description: The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.)
Assertion
Ref Expression
vnex ¬ ∃𝑥 𝑥 = V

Proof of Theorem vnex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nalset 4929 . 2 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 3354 . . . . . 6 𝑦 ∈ V
32tbt 358 . . . . 5 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1895 . . . 4 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2765 . . . 4 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 267 . . 3 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1924 . 2 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 311 1 ¬ ∃𝑥 𝑥 = V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1629   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915
This theorem depends on definitions:  df-bi 197  df-an 383  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353
This theorem is referenced by:  vprc  4931
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