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Theorem ruALT 4568
Description: Alternate proof of Russell's Paradox ru 2976, simplified using (indirectly) the Axiom of Set Induction ax-setind 4554. (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 4150 . . 3 ¬ V ∈ V
2 df-nel 2456 . . 3 (V ∉ V ↔ ¬ V ∈ V)
31, 2mpbir 146 . 2 V ∉ V
4 ruv 4567 . . 3 {𝑥𝑥𝑥} = V
5 neleq1 2459 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
64, 5ax-mp 5 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
73, 6mpbir 146 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 105   = wceq 1364  wcel 2160  {cab 2175  wnel 2455  Vcvv 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-v 2754  df-dif 3146  df-sn 3613
This theorem is referenced by: (None)
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