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

Theorem iunun 4362
 Description: Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunun

Proof of Theorem iunun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.43 2946 . . . 4
2 elun 3574 . . . . 5
32rexbii 2889 . . . 4
4 eliun 4283 . . . . 5
5 eliun 4283 . . . . 5
64, 5orbi12i 524 . . . 4
71, 3, 63bitr4i 281 . . 3
8 eliun 4283 . . 3
9 elun 3574 . . 3
107, 8, 93bitr4i 281 . 2
1110eqriv 2448 1
 Colors of variables: wff setvar class Syntax hints:   wo 370   wceq 1444   wcel 1887  wrex 2738   cun 3402  ciun 4278 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2742  df-rex 2743  df-v 3047  df-un 3409  df-iun 4280 This theorem is referenced by:  iununi  4366  oarec  7263  comppfsc  20547  uniiccdif  22535  bnj1415  29847  dftrpred4g  30475
 Copyright terms: Public domain W3C validator