Theorem ruv 9068

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		df-v 3498	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 2019	. . . 4 ⊢ 𝑥 = 𝑥
3		elirrv 9062	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
4	3	nelir 3128	. . . 4 ⊢ 𝑥 ∉ 𝑥
5	2, 4	2th 266	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝑥 ∉ 𝑥)
6	5	abbii 2888	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝑥 ∉ 𝑥}
7	1, 6	eqtr2i 2847	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1537  {cab 2801   ∉ wnel 3125  Vcvv 3496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-reg 9058

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-nel 3126  df-ral 3145  df-rex 3146  df-v 3498  df-dif 3941  df-un 3943  df-nul 4294  df-sn 4570  df-pr 4572

This theorem is referenced by:  ruALT  9069