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Theorem setind 4318
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 2999 . . . 4  |-  ( x 
C_  A  <->  A. y
( y  e.  x  ->  y  e.  A ) )
21imbi1i 236 . . 3  |-  ( ( x  C_  A  ->  x  e.  A )  <->  ( A. y ( y  e.  x  ->  y  e.  A )  ->  x  e.  A ) )
32albii 1400 . 2  |-  ( A. x ( x  C_  A  ->  x  e.  A
)  <->  A. x ( A. y ( y  e.  x  ->  y  e.  A )  ->  x  e.  A ) )
4 setindel 4317 . 2  |-  ( A. x ( A. y
( y  e.  x  ->  y  e.  A )  ->  x  e.  A
)  ->  A  =  _V )
53, 4sylbi 119 1  |-  ( A. x ( x  C_  A  ->  x  e.  A
)  ->  A  =  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283    = wceq 1285    e. wcel 1434   _Vcvv 2612    C_ wss 2984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-setind 4316
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-ral 2358  df-v 2614  df-in 2990  df-ss 2997
This theorem is referenced by:  setind2  4319
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