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Theorem iinpw 4054
 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 A = x A x
Distinct variable group:   x,A

Proof of Theorem iinpw
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 ssint 3942 . . . 4 (y Ax A y x)
2 vex 2862 . . . . . 6 y V
32elpw 3728 . . . . 5 (y xy x)
43ralbii 2638 . . . 4 (x A y xx A y x)
51, 4bitr4i 243 . . 3 (y Ax A y x)
62elpw 3728 . . 3 (y Ay A)
7 eliin 3974 . . . 4 (y V → (y x A xx A y x))
82, 7ax-mp 8 . . 3 (y x A xx A y x)
95, 6, 83bitr4i 268 . 2 (y Ay x A x)
109eqriv 2350 1 A = x A x
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∀wral 2614  Vcvv 2859   ⊆ wss 3257  ℘cpw 3722  ∩cint 3926  ∩ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724  df-int 3927  df-iin 3972 This theorem is referenced by: (None)
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