Theorem setind2 4572

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 3617	. 2 |- ( ~P A C_ A <-> A. x ( x C_ A -> x e. A ) )
2		setind 4571	. 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 1362   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   ~Pcpw 3601

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-v 2762  df-in 3159  df-ss 3166  df-pw 3603

This theorem is referenced by: (None)