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Theorem ruALT 8748
Description: Alternate proof of ru 3630, simplified using (indirectly) the Axiom of Regularity ax-reg 8737. (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 4990 . . 3 ¬ V ∈ V
21nelir 3075 . 2 V ∉ V
3 ruv 8747 . . 3 {𝑥𝑥𝑥} = V
4 neleq1 3077 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
53, 4ax-mp 5 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
62, 5mpbir 223 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  {cab 2783  wnel 3072  Vcvv 3383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095  ax-reg 8737
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-nel 3073  df-ral 3092  df-rex 3093  df-v 3385  df-dif 3770  df-un 3772  df-nul 4114  df-sn 4367  df-pr 4369
This theorem is referenced by: (None)
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