Theorem vprc 4136

Description: The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.)

Assertion

Ref	Expression
vprc	⊢ ¬ V ∈ V

Proof of Theorem vprc

Step	Hyp	Ref	Expression
1		vnex 4135	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		isset 2744	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
3	1, 2	mtbir 671	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2740

This theorem is referenced by:  nvel  4137  intexr  4151  intnexr  4152  abnex  4448  snnex  4449  ruALT  4551  dcextest  4581  iprc  4896  snexxph  6949  elfi2  6971  fi0  6974