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Theorem dfint3 4318
 Description: Alternate definition of class intersection for the existence proof. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
dfint3 A = ∼ ⋃1(kSkk A)

Proof of Theorem dfint3
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . . 7 x V
21eluni1 4173 . . . . . 6 (x 1(kSkk A) ↔ {x} (kSkk A))
3 snex 4111 . . . . . . 7 {x} V
43elimak 4259 . . . . . 6 ({x} (kSkk A) ↔ y Ay, {x}⟫ kSk )
52, 4bitri 240 . . . . 5 (x 1(kSkk A) ↔ y Ay, {x}⟫ kSk )
6 vex 2862 . . . . . . . 8 y V
76, 3opkelcnvk 4250 . . . . . . 7 (⟪y, {x}⟫ kSk ↔ ⟪{x}, ySk )
8 opkex 4113 . . . . . . . 8 ⟪{x}, y V
98elcompl 3225 . . . . . . 7 (⟪{x}, ySk ↔ ¬ ⟪{x}, y Sk )
101, 6elssetk 4270 . . . . . . . 8 (⟪{x}, y Skx y)
1110notbii 287 . . . . . . 7 (¬ ⟪{x}, y Sk ↔ ¬ x y)
127, 9, 113bitri 262 . . . . . 6 (⟪y, {x}⟫ kSk ↔ ¬ x y)
1312rexbii 2639 . . . . 5 (y Ay, {x}⟫ kSky A ¬ x y)
14 rexnal 2625 . . . . 5 (y A ¬ x y ↔ ¬ y A x y)
155, 13, 143bitri 262 . . . 4 (x 1(kSkk A) ↔ ¬ y A x y)
1615con2bii 322 . . 3 (y A x y ↔ ¬ x 1(kSkk A))
171elint2 3933 . . 3 (x Ay A x y)
181elcompl 3225 . . 3 (x ∼ ⋃1(kSkk A) ↔ ¬ x 1(kSkk A))
1916, 17, 183bitr4i 268 . 2 (x Ax ∼ ⋃1(kSkk A))
2019eqriv 2350 1 A = ∼ ⋃1(kSkk A)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615   ∼ ccompl 3205  {csn 3737  ∩cint 3926  ⟪copk 4057  ⋃1cuni1 4133  ◡kccnvk 4175   “k cimak 4179   Sk cssetk 4183 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-uni1 4138  df-cnvk 4186  df-imak 4189  df-ssetk 4193 This theorem is referenced by:  intexg  4319
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