Theorem setind 4538

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 3144	. . . 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 1470	. 2 |- ( A. x ( x C_ A -> x e. A ) <-> A. x ( A. y ( y e. x -> y e. A ) -> x e. A ) )
4		setindel 4537	. 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 1351   = wceq 1353   e. wcel 2148   _Vcvv 2737   C_ wss 3129

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4536

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-v 2739  df-in 3135  df-ss 3142

This theorem is referenced by:  setind2  4539