ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ruALT GIF version

Theorem ruALT 4303
Description: Alternate proof of Russell's Paradox ru 2786, simplified using (indirectly) the Axiom of Set Induction ax-setind 4290. (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 3916 . . 3 ¬ V ∈ V
2 df-nel 2315 . . 3 (V ∉ V ↔ ¬ V ∈ V)
31, 2mpbir 138 . 2 V ∉ V
4 ruv 4302 . . 3 {𝑥𝑥𝑥} = V
5 neleq1 2318 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
64, 5ax-mp 7 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
73, 6mpbir 138 1 {𝑥𝑥𝑥} ∉ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 102   = wceq 1259  wcel 1409  {cab 2042  wnel 2314  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-v 2576  df-dif 2948  df-sn 3409
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator