Theorem ruv 4274

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 2559	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
2		equid 1589	. . . 4 ⊢ 𝑥 = 𝑥
3		elirrv 4272	. . . . 5 ⊢ ¬ 𝑥 ∈ 𝑥
4	3	nelir 2300	. . . 4 ⊢ 𝑥 ∉ 𝑥
5	2, 4	2th 163	. . 3 ⊢ (𝑥 = 𝑥 ↔ 𝑥 ∉ 𝑥)
6	5	abbii 2153	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ 𝑥 ∉ 𝑥}
7	1, 6	eqtr2i 2061	1 ⊢ {𝑥 ∣ 𝑥 ∉ 𝑥} = V

Syntax hints:   = wceq 1243  {cab 2026   ∉ wnel 2205  Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-v 2559  df-dif 2920  df-sn 3381

This theorem is referenced by:  ruALT  4275