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Theorem ruALT 8391
Description: Alternate proof of ru 3401, simplified using (indirectly) the Axiom of Regularity ax-reg 8380. (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 4724 . . 3 ¬ V ∈ V
21nelir 2886 . 2 V ∉ V
3 ruv 8390 . . 3 {𝑥𝑥𝑥} = V
4 neleq1 2888 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
53, 4ax-mp 5 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
62, 5mpbir 220 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  {cab 2596  wnel 2781  Vcvv 3173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-reg 8380
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128
This theorem is referenced by: (None)
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