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Theorem setind 4537
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 3144 . . . 4  |-  ( x 
C_  A  <->  A. y
( y  e.  x  ->  y  e.  A ) )
21imbi1i 238 . . 3  |-  ( ( x  C_  A  ->  x  e.  A )  <->  ( A. y ( y  e.  x  ->  y  e.  A )  ->  x  e.  A ) )
32albii 1470 . 2  |-  ( A. x ( x  C_  A  ->  x  e.  A
)  <->  A. x ( A. y ( y  e.  x  ->  y  e.  A )  ->  x  e.  A ) )
4 setindel 4536 . 2  |-  ( A. x ( A. y
( y  e.  x  ->  y  e.  A )  ->  x  e.  A
)  ->  A  =  _V )
53, 4sylbi 121 1  |-  ( A. x ( x  C_  A  ->  x  e.  A
)  ->  A  =  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1351    = wceq 1353    e. wcel 2148   _Vcvv 2737    C_ wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4535
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142
This theorem is referenced by:  setind2  4538
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