Theorem ruv 9652

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 3476	. . . 4 ⊢ 𝑥 ∈ V
2		elirr 9647	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
3	2	nelir 3042	. . . 4 ⊢ 𝑥 ∉ 𝑥
4	1, 3	2th 263	. . 3 ⊢ (𝑥 ∈ V ↔ 𝑥 ∉ 𝑥)
5	4	eqabi 2865	. 2 ⊢ V = {𝑥 ∣ 𝑥 ∉ 𝑥}
6	5	eqcomi 2738	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1534   ∈ wcel 2100  {cab 2706   ∉ wnel 3039  Vcvv 3472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-12 2170  ax-ext 2700  ax-sep 5307  ax-pr 5437  ax-reg 9642

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-nel 3040  df-ral 3055  df-rex 3064  df-v 3474  df-un 3964  df-sn 4637  df-pr 4639

This theorem is referenced by:  ruALT  9653