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Theorem ruALT 9634
Description: Alternate proof of ru 3777, simplified using (indirectly) the Axiom of Regularity ax-reg 9623. (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 5319 . . 3 ¬ V ∈ V
21nelir 3046 . 2 V ∉ V
3 ruv 9633 . . 3 {𝑥𝑥𝑥} = V
4 neleq1 3049 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
53, 4ax-mp 5 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
62, 5mpbir 230 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  {cab 2705  wnel 3043  Vcvv 3473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-pr 5433  ax-reg 9623
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-nel 3044  df-ral 3059  df-rex 3068  df-v 3475  df-un 3954  df-sn 4633  df-pr 4635
This theorem is referenced by: (None)
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