Theorem vnex 5289

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.

Step	Hyp	Ref	Expression
1		nalset 5288	. 2 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2		vex 3468	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	tbt 369	. . . . 5 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
4	3	albii 1819	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
5		dfcleq 2729	. . . 4 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
6	4, 5	bitr4i 278	. . 3 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V)
7	6	exbii 1848	. 2 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V)
8	1, 7	mtbi 322	1 ⊢ ¬ ∃𝑥 𝑥 = V

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466

This theorem is referenced by:  vprc  5290