Theorem setind2 4517

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.

Step	Hyp	Ref	Expression
1		pwss 3575	. 2 |- ( ~P A C_ A <-> A. x ( x C_ A -> x e. A ) )
2		setind 4516	. 2 |- ( A. x ( x C_ A -> x e. A ) -> A = _V )
3	1, 2	sylbi 120	1 |- ( ~P A C_ A -> A = _V )

Syntax hints:   -> wi 4   A.wal 1341   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by: (None)