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Theorem dfuni3 4315
 Description: Alternate definition of class union for existence proof. (Contributed by SF, 14-Jan-2015.)
Assertion
Ref Expression
dfuni3 A = ⋃1(k Skk A)

Proof of Theorem dfuni3
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . 6 y V
2 snex 4111 . . . . . 6 {x} V
31, 2opkelcnvk 4250 . . . . 5 (⟪y, {x}⟫ k Sk ↔ ⟪{x}, y Sk )
4 vex 2862 . . . . . 6 x V
54, 1elssetk 4270 . . . . 5 (⟪{x}, y Skx y)
63, 5bitri 240 . . . 4 (⟪y, {x}⟫ k Skx y)
76rexbii 2639 . . 3 (y Ay, {x}⟫ k Sky A x y)
84eluni1 4173 . . . 4 (x 1(k Skk A) ↔ {x} (k Skk A))
92elimak 4259 . . . 4 ({x} (k Skk A) ↔ y Ay, {x}⟫ k Sk )
108, 9bitri 240 . . 3 (x 1(k Skk A) ↔ y Ay, {x}⟫ k Sk )
11 eluni2 3895 . . 3 (x Ay A x y)
127, 10, 113bitr4ri 269 . 2 (x Ax 1(k Skk A))
1312eqriv 2350 1 A = ⋃1(k Skk A)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {csn 3737  ∪cuni 3891  ⟪copk 4057  ⋃1cuni1 4133  ◡kccnvk 4175   “k cimak 4179   Sk cssetk 4183 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  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-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-opk 4058  df-1c 4136  df-uni1 4138  df-cnvk 4186  df-imak 4189  df-ssetk 4193 This theorem is referenced by:  uniexg  4316
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