Theorem vprc 4948

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

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nalset 4947	. . 3 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2		vex 3343	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	tbt 358	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
4	3	albii 1896	. . . . 5 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
5		dfcleq 2754	. . . . 5 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
6	4, 5	bitr4i 267	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V)
7	6	exbii 1923	. . 3 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V)
8	1, 7	mtbi 311	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
9		isset 3347	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
10	8, 9	mtbir 312	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1630   = wceq 1632  ∃wex 1853   ∈ wcel 2139  Vcvv 3340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933

This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1635  df-ex 1854  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-v 3342

This theorem is referenced by:  nvel  4949  vnex  4950  intex  4969  intnex  4970  abnex  7130  snnexOLD  7132  iprc  7266  opabn1stprc  7395  elfi2  8485  fi0  8491  ruALT  8673  cardmin2  9014  00lsp  19183  fveqvfvv  41710  ndmaovcl  41789