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

Theorem iunin1 4334
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4322 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 4333 . 2
2 incom 3616 . . . 4
32a1i 11 . . 3
43iuneq2i 4288 . 2
5 incom 3616 . 2
61, 4, 53eqtr4i 2503 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1452   wcel 1904   cin 3389  ciun 4269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-ss 3404  df-iun 4271 This theorem is referenced by:  2iunin  4337  tgrest  20252  metnrmlem3  21956  metnrmlem3OLD  21971  limciun  22928  uniin1  28242  disjunsn  28281  measinblem  29116  sstotbnd2  32170  sge0iunmptlemre  38371
 Copyright terms: Public domain W3C validator