ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ruv GIF version

Theorem ruv 4364
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 2621 . 2 V = {𝑥𝑥 = 𝑥}
2 equid 1634 . . . 4 𝑥 = 𝑥
3 elirrv 4362 . . . . 5 ¬ 𝑥𝑥
43nelir 2353 . . . 4 𝑥𝑥
52, 42th 172 . . 3 (𝑥 = 𝑥𝑥𝑥)
65abbii 2203 . 2 {𝑥𝑥 = 𝑥} = {𝑥𝑥𝑥}
71, 6eqtr2i 2109 1 {𝑥𝑥𝑥} = V
Colors of variables: wff set class
Syntax hints:   = wceq 1289  {cab 2074  wnel 2350  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-setind 4351
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-v 2621  df-dif 3001  df-sn 3450
This theorem is referenced by:  ruALT  4365
  Copyright terms: Public domain W3C validator