Theorem iinpw 3849

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 3734	. . . 4 |- ( y C_ |^| A <-> A. x e. A y C_ x )
2		vex 2644	. . . . . 6 |- y e. _V
3	2	elpw 3463	. . . . 5 |- ( y e. ~P x <-> y C_ x )
4	3	ralbii 2400	. . . 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 3463	. . 3 |- ( y e. ~P |^| A <-> y C_ |^| A )
7		eliin 3765	. . . 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 211	. 2 |- ( y e. ~P |^| A <-> y e. |^|_ x e. A ~P x )
10	9	eqriv 2097	1 |- ~P |^| A = |^|_ x e. A ~P x

Syntax hints:   <-> wb 104   = wceq 1299   e. wcel 1448   A.wral 2375   _Vcvv 2641   C_ wss 3021   ~Pcpw 3457   |^|cint 3718   |^|_ciin 3761

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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-int 3719  df-iin 3763

This theorem is referenced by: (None)