Theorem iinpw 4617

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 4493	. . . 4 |- ( y C_ |^| A <-> A. x e. A y C_ x )
2		selpw 4165	. . . . 5 |- ( y e. ~P x <-> y C_ x )
3	2	ralbii 2980	. . . 4 |- ( A. x e. A y e. ~P x <-> A. x e. A y C_ x )
4	1, 3	bitr4i 267	. . 3 |- ( y C_ |^| A <-> A. x e. A y e. ~P x )
5		selpw 4165	. . 3 |- ( y e. ~P |^| A <-> y C_ |^| A )
6		vex 3203	. . . 4 |- y e. _V
7		eliin 4525	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A ~P x <-> A. x e. A y e. ~P x ) )
8	6, 7	ax-mp 5	. . 3 |- ( y e. |^|_ x e. A ~P x <-> A. x e. A y e. ~P x )
9	4, 5, 8	3bitr4i 292	. 2 |- ( y e. ~P |^| A <-> y e. |^|_ x e. A ~P x )
10	9	eqriv 2619	1 |- ~P |^| A = |^|_ x e. A ~P x

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-int 4476  df-iin 4523

This theorem is referenced by: (None)