Theorem setind 4383

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 3028	. . . 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 1411	. 2 |- ( A. x ( x C_ A -> x e. A ) <-> A. x ( A. y ( y e. x -> y e. A ) -> x e. A ) )
4		setindel 4382	. 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 1294   = wceq 1296   e. wcel 1445   _Vcvv 2633   C_ wss 3013

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-ral 2375  df-v 2635  df-in 3019  df-ss 3026

This theorem is referenced by:  setind2  4384