Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunin1 Unicode version

Theorem iunin1 4120
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4108 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
iunin1
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem iunin1
StepHypRef Expression
1 iunin2 4119 . 2
2 incom 3497 . . . 4
32a1i 11 . . 3
43iuneq2i 4075 . 2
5 incom 3497 . 2
61, 4, 53eqtr4i 2438 1
 Colors of variables: wff set class Syntax hints:   wceq 1649   wcel 1721   cin 3283  ciun 4057 This theorem is referenced by:  2iunin  4123  tgrest  17181  metnrmlem3  18848  limciun  19738  measinblem  24531  sstotbnd2  26377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389 This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-v 2922  df-in 3291  df-ss 3298  df-iun 4059
 Copyright terms: Public domain W3C validator