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Theorem setind 4454
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 3086 . . . 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 1446 . 2  |-  ( A. x ( x  C_  A  ->  x  e.  A
)  <->  A. x ( A. y ( y  e.  x  ->  y  e.  A )  ->  x  e.  A ) )
4 setindel 4453 . 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 1329    = wceq 1331    e. wcel 1480   _Vcvv 2686    C_ wss 3071
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084
This theorem is referenced by:  setind2  4455
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