Theorem ruv 9556

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 3458	. . . 4 ⊢ 𝑥 ∈ V
2		elirr 9548	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3064	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 266	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2897	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2771	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1560   ∈ wcel 2142  {cab 2740   ∉ wnel 3061  Vcvv 3454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-reg 9540

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nel 3062  df-v 3456

This theorem is referenced by:  ruALT  9557