Theorem setind 4571

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		dfss2 3168	. . . 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 1481	. 2 |- ( A. x ( x C_ A -> x e. A ) <-> A. x ( A. y ( y e. x -> y e. A ) -> x e. A ) )
4		setindel 4570	. 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 1362   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153

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

This theorem is referenced by:  setind2  4572