Theorem ruv 9196

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 3402	. . . 4 ⊢ 𝑥 ∈ V
2		elirr 9191	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3039	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 267	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	abbi2i 2869	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2745	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1543   ∈ wcel 2112  {cab 2714   ∉ wnel 3036  Vcvv 3398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-reg 9186

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-nel 3037  df-ral 3056  df-rex 3057  df-v 3400  df-dif 3856  df-un 3858  df-nul 4224  df-sn 4528  df-pr 4530

This theorem is referenced by:  ruALT  9197