Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunin2 Unicode version

Theorem iunin2 3864
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3853 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
iunin2
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem iunin2
StepHypRef Expression
1 r19.42v 2656 . . . 4
2 elin 3266 . . . . 5
32rexbii 2532 . . . 4
4 eliun 3807 . . . . 5
54anbi2i 678 . . . 4
61, 3, 53bitr4i 270 . . 3
7 eliun 3807 . . 3
8 elin 3266 . . 3
96, 7, 83bitr4i 270 . 2
109eqriv 2250 1
 Colors of variables: wff set class Syntax hints:   wa 360   wceq 1619   wcel 1621  wrex 2510   cin 3077  ciun 3803 This theorem is referenced by:  iunin1  3865  2iunin  3868  resiundiOLD  4652  resiun1  4881  resiun2  4882  kmlem11  7670  cmpsublem  16958  cmpsub  16959  kgentopon  17065  metnrmlem3  18197  ovoliunlem1  18693  voliunlem1  18739  voliunlem2  18740  uniioombllem2  18770  uniioombllem4  18773  volsup2  18792  itg1addlem5  18887  itg1climres  18901  cvmscld  22975  isunscov  24239  heiborlem3  25703 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-rex 2514  df-v 2729  df-in 3085  df-iun 3805
 Copyright terms: Public domain W3C validator