Theorem ruv 9555

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 3451	. . . 4 ⊢ 𝑥 ∈ V
2		elirr 9550	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3032	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 264	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2863	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2738	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1540   ∈ wcel 2109  {cab 2707   ∉ wnel 3029  Vcvv 3447

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-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-pr 5387  ax-reg 9545

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nel 3030  df-ral 3045  df-rex 3054  df-v 3449  df-un 3919  df-sn 4590  df-pr 4592

This theorem is referenced by:  ruALT  9556