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Theorem setind2 4539
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 A  C_  A  ->  A  =  _V )

Proof of Theorem setind2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 pwss 3591 . 2  |-  ( ~P A  C_  A  <->  A. x
( x  C_  A  ->  x  e.  A ) )
2 setind 4538 . 2  |-  ( A. x ( x  C_  A  ->  x  e.  A
)  ->  A  =  _V )
31, 2sylbi 121 1  |-  ( ~P A  C_  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   ~Pcpw 3575
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 4536
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  df-pw 3577
This theorem is referenced by: (None)
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