Theorem ruv 9639

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 9631	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3068	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 267	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2899	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2772	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  {cab 2741   ∉ wnel 3065  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-reg 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nel 3066  df-v 3456

This theorem is used by:  ruALT  9640