Theorem vprc 4724

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 4723	. . 3 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	tbt 358	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
4	3	albii 1737	. . . . 5 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
5		dfcleq 2604	. . . . 5 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
6	4, 5	bitr4i 266	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V)
7	6	exbii 1764	. . 3 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V)
8	1, 7	mtbi 311	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
9		isset 3180	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
10	8, 9	mtbir 312	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709

This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175

This theorem is referenced by:  nvel  4725  vnex  4726  intex  4747  intnex  4748  snnex  6862  iprc  6993  elfi2  8203  fi0  8209  ruALT  8391  cardmin2  8707  00lsp  18802  bj-xnex  32245  fveqvfvv  39853  ndmaovcl  39932  opabn1stprc  40318