Theorem ruv 9566

Description: The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)

Assertion

Ref	Expression
ruv	⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Proof of Theorem ruv

Step	Hyp	Ref	Expression
1		vex 3459	. . . 4 ⊢ 𝑥 ∈ V
2		elirr 9558	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3067	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 267	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2898	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2772	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1570   ∈ wcel 2143  {cab 2741   ∉ wnel 3064  Vcvv 3455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-reg 9550

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nel 3065  df-v 3457

This theorem is referenced by:  ruALT  9567