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Theorem ruALT 4461
Description: Alternate proof of Russell's Paradox ru 2903, simplified using (indirectly) the Axiom of Set Induction ax-setind 4447. (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 4055 . . 3 ¬ V ∈ V
2 df-nel 2402 . . 3 (V ∉ V ↔ ¬ V ∈ V)
31, 2mpbir 145 . 2 V ∉ V
4 ruv 4460 . . 3 {𝑥𝑥𝑥} = V
5 neleq1 2405 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
64, 5ax-mp 5 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
73, 6mpbir 145 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 104   = wceq 1331  wcel 1480  {cab 2123  wnel 2401  Vcvv 2681
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-v 2683  df-dif 3068  df-sn 3528
This theorem is referenced by: (None)
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