Theorem iinpw 3939

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 3823	. . . 4 |- ( y C_ |^| A <-> A. x e. A y C_ x )
2		vex 2715	. . . . . 6 |- y e. _V
3	2	elpw 3549	. . . . 5 |- ( y e. ~P x <-> y C_ x )
4	3	ralbii 2463	. . . 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 3549	. . 3 |- ( y e. ~P |^| A <-> y C_ |^| A )
7		eliin 3854	. . . 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 2154	1 |- ~P |^| A = |^|_ x e. A ~P x

Syntax hints:   <-> wb 104   = wceq 1335   e. wcel 2128   A.wral 2435   _Vcvv 2712   C_ wss 3102   ~Pcpw 3543   |^|cint 3807   |^|_ciin 3850

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545  df-int 3808  df-iin 3852

This theorem is referenced by: (None)