Theorem setind 4585

Description: Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)

Assertion

Ref	Expression
setind	|- ( A. x ( x C_ A -> x e. A ) -> A = _V )

Distinct variable group:   x, A

Proof of Theorem setind

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssalel 3180	. . . 4 |- ( x C_ A <-> A. y ( y e. x -> y e. A ) )
2	1	imbi1i 238	. . 3 |- ( ( x C_ A -> x e. A ) <-> ( A. y ( y e. x -> y e. A ) -> x e. A ) )
3	2	albii 1492	. 2 |- ( A. x ( x C_ A -> x e. A ) <-> A. x ( A. y ( y e. x -> y e. A ) -> x e. A ) )
4		setindel 4584	. 2 |- ( A. x ( A. y ( y e. x -> y e. A ) -> x e. A ) -> A = _V )
5	3, 4	sylbi 121	1 |- ( A. x ( x C_ A -> x e. A ) -> A = _V )

Syntax hints:   -> wi 4   A.wal 1370   = wceq 1372   e. wcel 2175   _Vcvv 2771   C_ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186  ax-setind 4583

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-ral 2488  df-v 2773  df-in 3171  df-ss 3178

This theorem is referenced by:  setind2  4586