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Theorem ruALT 9143
Description: Alternate proof of ru 3680, simplified using (indirectly) the Axiom of Regularity ax-reg 9132. (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 5184 . . 3 ¬ V ∈ V
21nelir 3042 . 2 V ∉ V
3 ruv 9142 . . 3 {𝑥𝑥𝑥} = V
4 neleq1 3044 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
53, 4ax-mp 5 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
62, 5mpbir 234 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1542  {cab 2717  wnel 3039  Vcvv 3399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297  ax-reg 9132
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nel 3040  df-ral 3059  df-rex 3060  df-v 3401  df-dif 3847  df-un 3849  df-nul 4213  df-sn 4518  df-pr 4520
This theorem is referenced by: (None)
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