Theorem setind2 4450

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 3521	. 2 |- ( ~P A C_ A <-> A. x ( x C_ A -> x e. A ) )
2		setind 4449	. 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 1329   = wceq 1331   e. wcel 1480   _Vcvv 2681   C_ wss 3066   ~Pcpw 3505

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-ral 2419  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507

This theorem is referenced by: (None)