Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  vnex GIF version

Theorem vnex 4019
 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 4018 . 2 ¬ ∃𝑥𝑦 𝑦𝑥
2 vex 2660 . . . . . 6 𝑦 ∈ V
32tbt 246 . . . . 5 (𝑦𝑥 ↔ (𝑦𝑥𝑦 ∈ V))
43albii 1429 . . . 4 (∀𝑦 𝑦𝑥 ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
5 dfcleq 2109 . . . 4 (𝑥 = V ↔ ∀𝑦(𝑦𝑥𝑦 ∈ V))
64, 5bitr4i 186 . . 3 (∀𝑦 𝑦𝑥𝑥 = V)
76exbii 1567 . 2 (∃𝑥𝑦 𝑦𝑥 ↔ ∃𝑥 𝑥 = V)
81, 7mtbi 642 1 ¬ ∃𝑥 𝑥 = V
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 104  ∀wal 1312   = wceq 1314  ∃wex 1451   ∈ wcel 1463  Vcvv 2657 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097  ax-sep 4006 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2659 This theorem is referenced by:  vprc  4020
 Copyright terms: Public domain W3C validator