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

Theorem iuncom 4306
 Description: Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
iuncom
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   (,)

Proof of Theorem iuncom
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 rexcom 2987 . . . 4
2 eliun 4304 . . . . 5
32rexbii 2924 . . . 4
4 eliun 4304 . . . . 5
54rexbii 2924 . . . 4
61, 3, 53bitr4i 280 . . 3
7 eliun 4304 . . 3
8 eliun 4304 . . 3
96, 7, 83bitr4i 280 . 2
109eqriv 2418 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1437   wcel 1872  wrex 2772  ciun 4299 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ral 2776  df-rex 2777  df-v 3082  df-iun 4301 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator