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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 ∼ ⋃1kSk k

Proof of Theorem dfint3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2862 . . . . . . 7
21eluni1 4173 . . . . . 6 1kSk k kSk k
3 snex 4111 . . . . . . 7
43elimak 4259 . . . . . 6 kSk k kSk
52, 4bitri 240 . . . . 5 1kSk k kSk
6 vex 2862 . . . . . . . 8
76, 3opkelcnvk 4250 . . . . . . 7 kSk Sk
8 opkex 4113 . . . . . . . 8
98elcompl 3225 . . . . . . 7 Sk Sk
101, 6elssetk 4270 . . . . . . . 8 Sk
1110notbii 287 . . . . . . 7 Sk
127, 9, 113bitri 262 . . . . . 6 kSk
1312rexbii 2639 . . . . 5 kSk
14 rexnal 2625 . . . . 5
155, 13, 143bitri 262 . . . 4 1kSk k
1615con2bii 322 . . 3 1kSk k
171elint2 3933 . . 3
181elcompl 3225 . . 3 ∼ ⋃1kSk k 1kSk k
1916, 17, 183bitr4i 268 . 2 ∼ ⋃1kSk k
2019eqriv 2350 1 ∼ ⋃1kSk k
 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  kcimak 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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