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Theorem ruALT 9059
 Description: Alternate proof of ru 3769, simplified using (indirectly) the Axiom of Regularity ax-reg 9048. (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 5210 . . 3 ¬ V ∈ V
21nelir 3124 . 2 V ∉ V
3 ruv 9058 . . 3 {𝑥𝑥𝑥} = V
4 neleq1 3126 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
53, 4ax-mp 5 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
62, 5mpbir 233 1 {𝑥𝑥𝑥} ∉ V
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1530  {cab 2797   ∉ wnel 3121  Vcvv 3493 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-reg 9048 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-nel 3122  df-ral 3141  df-rex 3142  df-v 3495  df-dif 3937  df-un 3939  df-nul 4290  df-sn 4560  df-pr 4562 This theorem is referenced by: (None)
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