Theorem iinpw 3733

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 ~Px

Distinct variable group:   x,A

Proof of Theorem iinpw

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 3622	. . . 4 |- (y C_ |^|A <-> A.x e. A y C_ x)
2		vex 2554	. . . . . 6 |- y e. _V
3	2	elpw 3357	. . . . 5 |- (y e. ~Px <-> y C_ x)
4	3	ralbii 2324	. . . 4 |- (A.x e. A y e. ~Px <-> A.x e. A y C_ x)
5	1, 4	bitr4i 176	. . 3 |- (y C_ |^|A <-> A.x e. A y e. ~Px)
6	2	elpw 3357	. . 3 |- (y e. ~P |^|A <-> y C_ |^|A)
7		eliin 3653	. . . 4 |- (y e. _V -> (y e. |^|_x e. A ~Px <-> A.x e. A y e. ~Px))
8	2, 7	ax-mp 7	. . 3 |- (y e. |^|_x e. A ~Px <-> A.x e. A y e. ~Px)
9	5, 6, 8	3bitr4i 201	. 2 |- (y e. ~P |^|A <-> y e. |^|_x e. A ~Px)
10	9	eqriv 2034	1 |- ~P |^|A = |^|_x e. A ~Px

Syntax hints:   <-> wb 98   = wceq 1242   e. wcel 1390  A.wral 2300   _Vcvv 2551   C_ wss 2911   ~Pcpw 3351   |^|cint 3606   |^|_ciin 3649

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-int 3607  df-iin 3651

This theorem is referenced by: (None)