MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ruALT Structured version   Visualization version   GIF version

Theorem ruALT 8460
Description: Alternate proof of ru 3420, simplified using (indirectly) the Axiom of Regularity ax-reg 8449. (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 4761 . . 3 ¬ V ∈ V
21nelir 2896 . 2 V ∉ V
3 ruv 8459 . . 3 {𝑥𝑥𝑥} = V
4 neleq1 2898 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
53, 4ax-mp 5 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
62, 5mpbir 221 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  {cab 2607  wnel 2893  Vcvv 3189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-reg 8449
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-nel 2894  df-ral 2912  df-rex 2913  df-v 3191  df-dif 3562  df-un 3564  df-nul 3897  df-sn 4154  df-pr 4156
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator