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Theorem vnex 3962
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 3961 . 2 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 2622 . . . . . 6 𝑦 ∈ V
32tbt 245 . . . . 5 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1404 . . . 4 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2082 . . . 4 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 185 . . 3 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1541 . 2 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 630 1 ¬ ∃𝑥 𝑥 = V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070  ax-sep 3949
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  vprc  3963
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