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

Theorem iuniin 3813
 Description: Law combining indexed union with indexed intersection. Eq. 14 in in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iuniin
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   (,)

Proof of Theorem iuniin
StepHypRef Expression
1 r19.12 2618 . . . 4
2 vex 2730 . . . . . 6
3 eliin 3808 . . . . . 6
42, 3ax-mp 10 . . . . 5
54rexbii 2532 . . . 4
6 eliun 3807 . . . . 5
76ralbii 2531 . . . 4
81, 5, 73imtr4i 259 . . 3
9 eliun 3807 . . 3
10 eliin 3808 . . . 4
112, 10ax-mp 10 . . 3
128, 9, 113imtr4i 259 . 2
1312ssriv 3105 1
 Colors of variables: wff set class Syntax hints:   wb 178   wcel 1621  wral 2509  wrex 2510  cvv 2727   wss 3078  ciun 3803  ciin 3804 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-ss 3089  df-iun 3805  df-iin 3806
 Copyright terms: Public domain W3C validator