Theorem setind2 4682

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 3704	. 2 |- ( ~P A C_ A <-> A. x ( x C_ A -> x e. A ) )
2		setind 4681	. 2 |- ( A. x ( x C_ A -> x e. A ) -> A = _V )
3	1, 2	sylbi 121	1 |- ( ~P A C_ A -> A = _V )

Syntax hints:   -> wi 4   A.wal 1400   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ~Pcpw 3685

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687

This theorem is referenced by: (None)