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

Theorem setind 4424
Description: Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
Assertion
Ref Expression
setind  |-  ( A. x ( x  C_  A  ->  x  e.  A
)  ->  A  =  _V )
Distinct variable group:    x, A

Proof of Theorem setind
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfss2 3056 . . . 4  |-  ( x 
C_  A  <->  A. y
( y  e.  x  ->  y  e.  A ) )
21imbi1i 237 . . 3  |-  ( ( x  C_  A  ->  x  e.  A )  <->  ( A. y ( y  e.  x  ->  y  e.  A )  ->  x  e.  A ) )
32albii 1431 . 2  |-  ( A. x ( x  C_  A  ->  x  e.  A
)  <->  A. x ( A. y ( y  e.  x  ->  y  e.  A )  ->  x  e.  A ) )
4 setindel 4423 . 2  |-  ( A. x ( A. y
( y  e.  x  ->  y  e.  A )  ->  x  e.  A
)  ->  A  =  _V )
53, 4sylbi 120 1  |-  ( A. x ( x  C_  A  ->  x  e.  A
)  ->  A  =  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1314    = wceq 1316    e. wcel 1465   _Vcvv 2660    C_ wss 3041
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
This theorem is referenced by:  setind2  4425
  Copyright terms: Public domain W3C validator