Theorem ruv 4364

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 2621	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 1634	. . . 4 ⊢ 𝑥 = 𝑥
3		elirrv 4362	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
4	3	nelir 2353	. . . 4 ⊢ 𝑥 ∉ 𝑥
5	2, 4	2th 172	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝑥 ∉ 𝑥)
6	5	abbii 2203	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝑥 ∉ 𝑥}
7	1, 6	eqtr2i 2109	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1289  {cab 2074   ∉ wnel 2350  Vcvv 2619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4351

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-v 2621  df-dif 3001  df-sn 3450

This theorem is referenced by:  ruALT  4365