Theorem setind2 4369

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 3449	. 2 |- ( ~P A C_ A <-> A. x ( x C_ A -> x e. A ) )
2		setind 4368	. 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 1288   = wceq 1290   e. wcel 1439   _Vcvv 2620   C_ wss 3000   ~Pcpw 3433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-setind 4366

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-ral 2365  df-v 2622  df-in 3006  df-ss 3013  df-pw 3435

This theorem is referenced by: (None)