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Theorem ruALT 9547
Description: Alternate proof of ru 3742, simplified using (indirectly) the Axiom of Regularity ax-reg 9536. (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 5276 . . 3 ¬ V ∈ V
21nelir 3049 . 2 V ∉ V
3 ruv 9546 . . 3 {𝑥𝑥𝑥} = V
4 neleq1 3051 . . 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 1542  {cab 2710  wnel 3046  Vcvv 3447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-pr 5388  ax-reg 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nel 3047  df-ral 3062  df-rex 3071  df-v 3449  df-un 3919  df-sn 4591  df-pr 4593
This theorem is referenced by: (None)
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