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Theorem ruv 4511
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 2714 . 2 V = {𝑥𝑥 = 𝑥}
2 equid 1681 . . . 4 𝑥 = 𝑥
3 elirrv 4509 . . . . 5 ¬ 𝑥𝑥
43nelir 2425 . . . 4 𝑥𝑥
52, 42th 173 . . 3 (𝑥 = 𝑥𝑥𝑥)
65abbii 2273 . 2 {𝑥𝑥 = 𝑥} = {𝑥𝑥𝑥}
71, 6eqtr2i 2179 1 {𝑥𝑥𝑥} = V
Colors of variables: wff set class
Syntax hints:   = wceq 1335  {cab 2143  wnel 2422  Vcvv 2712
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-setind 4498
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-v 2714  df-dif 3104  df-sn 3567
This theorem is referenced by:  ruALT  4512
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