ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  setind2 GIF version

Theorem setind2 4425
Description: Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
setind2 (𝒫 𝐴𝐴𝐴 = V)

Proof of Theorem setind2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pwss 3496 . 2 (𝒫 𝐴𝐴 ↔ ∀𝑥(𝑥𝐴𝑥𝐴))
2 setind 4424 . 2 (∀𝑥(𝑥𝐴𝑥𝐴) → 𝐴 = V)
31, 2sylbi 120 1 (𝒫 𝐴𝐴𝐴 = V)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314   = wceq 1316  wcel 1465  Vcvv 2660  wss 3041  𝒫 cpw 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-ral 2398  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator