Theorem setind 4462

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 3091	. . . 4 |- ( x C_ A <-> A. y ( y e. x -> y e. A ) )
2	1	imbi1i 237	. . 3 |- ( ( x C_ A -> x e. A ) <-> ( A. y ( y e. x -> y e. A ) -> x e. A ) )
3	2	albii 1447	. 2 |- ( A. x ( x C_ A -> x e. A ) <-> A. x ( A. y ( y e. x -> y e. A ) -> x e. A ) )
4		setindel 4461	. 2 |- ( A. x ( A. y ( y e. x -> y e. A ) -> x e. A ) -> A = _V )
5	3, 4	sylbi 120	1 |- ( A. x ( x C_ A -> x e. A ) -> A = _V )

Syntax hints:   -> wi 4   A.wal 1330   = wceq 1332   e. wcel 1481   _Vcvv 2689   C_ wss 3076

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-v 2691  df-in 3082  df-ss 3089

This theorem is referenced by:  setind2  4463