Theorem iinpw 3963

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 3847	. . . 4 |- ( y C_ |^| A <-> A. x e. A y C_ x )
2		vex 2733	. . . . . 6 |- y e. _V
3	2	elpw 3572	. . . . 5 |- ( y e. ~P x <-> y C_ x )
4	3	ralbii 2476	. . . 4 |- ( A. x e. A y e. ~P x <-> A. x e. A y C_ x )
5	1, 4	bitr4i 186	. . 3 |- ( y C_ |^| A <-> A. x e. A y e. ~P x )
6	2	elpw 3572	. . 3 |- ( y e. ~P |^| A <-> y C_ |^| A )
7		eliin 3878	. . . 4 |- ( y e. _V -> ( y e. |^|_ x e. A ~P x <-> A. x e. A y e. ~P x ) )
8	2, 7	ax-mp 5	. . 3 |- ( y e. |^|_ x e. A ~P x <-> A. x e. A y e. ~P x )
9	5, 6, 8	3bitr4i 211	. 2 |- ( y e. ~P |^| A <-> y e. |^|_ x e. A ~P x )
10	9	eqriv 2167	1 |- ~P |^| A = |^|_ x e. A ~P x

Syntax hints:   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566   |^|cint 3831   |^|_ciin 3874

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-int 3832  df-iin 3876

This theorem is referenced by: (None)