Theorem ruv 4641

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 2801	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 1747	. . . 4 ⊢ 𝑥 = 𝑥
3		elirrv 4639	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
4	3	nelir 2498	. . . 4 ⊢ 𝑥 ∉ 𝑥
5	2, 4	2th 174	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝑥 ∉ 𝑥)
6	5	abbii 2345	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝑥 ∉ 𝑥}
7	1, 6	eqtr2i 2251	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1395  {cab 2215   ∉ wnel 2495  Vcvv 2799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-v 2801  df-dif 3199  df-sn 3672

This theorem is referenced by:  ruALT  4642