Theorem iinpw 3989

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 3872	. . . 4 |- ( y C_ |^| A <-> A. x e. A y C_ x )
2		vex 2752	. . . . . 6 |- y e. _V
3	2	elpw 3593	. . . . 5 |- ( y e. ~P x <-> y C_ x )
4	3	ralbii 2493	. . . 4 |- ( A. x e. A y e. ~P x <-> A. x e. A y C_ x )
5	1, 4	bitr4i 187	. . 3 |- ( y C_ |^| A <-> A. x e. A y e. ~P x )
6	2	elpw 3593	. . 3 |- ( y e. ~P |^| A <-> y C_ |^| A )
7		eliin 3903	. . . 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 212	. 2 |- ( y e. ~P |^| A <-> y e. |^|_ x e. A ~P x )
10	9	eqriv 2184	1 |- ~P |^| A = |^|_ x e. A ~P x

Syntax hints:   <-> wb 105   = wceq 1363   e. wcel 2158   A.wral 2465   _Vcvv 2749   C_ wss 3141   ~Pcpw 3587   |^|cint 3856   |^|_ciin 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-in 3147  df-ss 3154  df-pw 3589  df-int 3857  df-iin 3901

This theorem is referenced by: (None)