Theorem ruv 4527

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 2728	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 1689	. . . 4 ⊢ 𝑥 = 𝑥
3		elirrv 4525	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
4	3	nelir 2434	. . . 4 ⊢ 𝑥 ∉ 𝑥
5	2, 4	2th 173	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝑥 ∉ 𝑥)
6	5	abbii 2282	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝑥 ∉ 𝑥}
7	1, 6	eqtr2i 2187	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1343  {cab 2151   ∉ wnel 2431  Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-v 2728  df-dif 3118  df-sn 3582

This theorem is referenced by:  ruALT  4528