Theorem vnex 5220

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 5219	. 2 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2		vex 3499	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	tbt 372	. . . . 5 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
4	3	albii 1820	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
5		dfcleq 2817	. . . 4 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
6	4, 5	bitr4i 280	. . 3 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V)
7	6	exbii 1848	. 2 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V)
8	1, 7	mtbi 324	1 ⊢ ¬ ∃𝑥 𝑥 = V

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2795  ax-sep 5205

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-v 3498

This theorem is referenced by:  vprc  5221