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Theorem ruv 4302
 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 {𝑥𝑥𝑥} = V

Proof of Theorem ruv
StepHypRef Expression
1 df-v 2576 . 2 V = {𝑥𝑥 = 𝑥}
2 equid 1605 . . . 4 𝑥 = 𝑥
3 elirrv 4300 . . . . 5 ¬ 𝑥𝑥
43nelir 2317 . . . 4 𝑥𝑥
52, 42th 167 . . 3 (𝑥 = 𝑥𝑥𝑥)
65abbii 2169 . 2 {𝑥𝑥 = 𝑥} = {𝑥𝑥𝑥}
71, 6eqtr2i 2077 1 {𝑥𝑥𝑥} = V
 Colors of variables: wff set class Syntax hints:   = wceq 1259  {cab 2042   ∉ wnel 2314  Vcvv 2574 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-v 2576  df-dif 2948  df-sn 3409 This theorem is referenced by:  ruALT  4303
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