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Theorem ruALT 4357
Description: Alternate proof of Russell's Paradox ru 2837, simplified using (indirectly) the Axiom of Set Induction ax-setind 4343. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ruALT {𝑥𝑥𝑥} ∉ V

Proof of Theorem ruALT
StepHypRef Expression
1 vprc 3963 . . 3 ¬ V ∈ V
2 df-nel 2351 . . 3 (V ∉ V ↔ ¬ V ∈ V)
31, 2mpbir 144 . 2 V ∉ V
4 ruv 4356 . . 3 {𝑥𝑥𝑥} = V
5 neleq1 2354 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
64, 5ax-mp 7 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
73, 6mpbir 144 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 103   = wceq 1289  wcel 1438  {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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  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 2999  df-sn 3447
This theorem is referenced by: (None)
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