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

Theorem ruALT 4275
 Description: Alternate proof of Russell's Paradox ru 2763, simplified using (indirectly) the Axiom of Set Induction ax-setind 4262. (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 3888 . . 3 ¬ V ∈ V
2 df-nel 2207 . . 3 (V ∉ V ↔ ¬ V ∈ V)
31, 2mpbir 134 . 2 V ∉ V
4 ruv 4274 . . 3 {𝑥𝑥𝑥} = V
5 neleq1 2301 . . 3 ({𝑥𝑥𝑥} = V → ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V))
64, 5ax-mp 7 . 2 ({𝑥𝑥𝑥} ∉ V ↔ V ∉ V)
73, 6mpbir 134 1 {𝑥𝑥𝑥} ∉ V
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026   ∉ wnel 2205  Vcvv 2557 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-v 2559  df-dif 2920  df-sn 3381 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator