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Theorem vnex 4988
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 4987 . 2 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 3393 . . . . . 6 𝑦 ∈ V
32tbt 360 . . . . 5 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1907 . . . 4 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2799 . . . 4 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 269 . . 3 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1936 . 2 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 313 1 ¬ ∃𝑥 𝑥 = V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wal 1635   = wceq 1637  wex 1859  wcel 2158  Vcvv 3390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971
This theorem depends on definitions:  df-bi 198  df-an 385  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-clab 2792  df-cleq 2798  df-clel 2801  df-v 3392
This theorem is referenced by:  vprc  4989
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