Theorem vprc 4796

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 4795	. . 3 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2		vex 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
3	2	tbt 359	. . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
4	3	albii 1747	. . . . 5 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
5		dfcleq 2616	. . . . 5 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
6	4, 5	bitr4i 267	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V)
7	6	exbii 1774	. . 3 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V)
8	1, 7	mtbi 312	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
9		isset 3207	. 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V)
10	8, 9	mtbir 313	1 ⊢ ¬ V ∈ V

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  nvel  4797  vnex  4798  intex  4820  intnex  4821  abnex  6965  snnexOLD  6967  iprc  7101  opabn1stprc  7228  elfi2  8320  fi0  8326  ruALT  8508  cardmin2  8824  00lsp  18981  fveqvfvv  41204  ndmaovcl  41283