Theorem iinpw 3763

Description: The power class of an intersection in terms of indexed intersection. Exercise 24(a) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)

Assertion

Ref	Expression
iinpw	|- ~P |^| A = |^|_ x e. A ~P x

Distinct variable group:   x, A

Proof of Theorem iinpw

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 3652	. . . 4 |- ( y C_ |^| A <-> A. x e. A y C_ x )
2		vex 2604	. . . . . 6 |- y e. _V
3	2	elpw 3388	. . . . 5 |- ( y e. ~P x <-> y C_ x )
4	3	ralbii 2372	. . . 4 |- ( A. x e. A y e. ~P x <-> A. x e. A y C_ x )
5	1, 4	bitr4i 185	. . 3 |- ( y C_ |^| A <-> A. x e. A y e. ~P x )
6	2	elpw 3388	. . 3 |- ( y e. ~P |^| A <-> y C_ |^| A )
7		eliin 3683	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A ~P x <-> A. x e. A y e. ~P x ) )
8	2, 7	ax-mp 7	. . 3 |- ( y e. |^|_ x e. A ~P x <-> A. x e. A y e. ~P x )
9	5, 6, 8	3bitr4i 210	. 2 |- ( y e. ~P |^| A <-> y e. |^|_ x e. A ~P x )
10	9	eqriv 2078	1 |- ~P |^| A = |^|_ x e. A ~P x

Syntax hints:   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   _Vcvv 2601   C_ wss 2973   ~Pcpw 3382   |^|cint 3636   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-int 3637  df-iin 3681

This theorem is referenced by: (None)