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Theorem vnex 5185
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 5184 . 2 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 3447 . . . . . 6 𝑦 ∈ V
32tbt 373 . . . . 5 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1821 . . . 4 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2795 . . . 4 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 281 . . 3 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1849 . 2 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 325 1 ¬ ∃𝑥 𝑥 = V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wal 1536   = wceq 1538  wex 1781  wcel 2112  Vcvv 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773  ax-sep 5170
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446
This theorem is referenced by:  vprc  5186
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