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Theorem iuniin 3823
 Description: Law combining indexed union with indexed intersection. Eq. 14 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
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 r19.12 2538 . . . 4
2 vex 2689 . . . . . 6
3 eliin 3818 . . . . . 6
42, 3ax-mp 5 . . . . 5
54rexbii 2442 . . . 4
6 eliun 3817 . . . . 5
76ralbii 2441 . . . 4
81, 5, 73imtr4i 200 . . 3
9 eliun 3817 . . 3
10 eliin 3818 . . . 4
112, 10ax-mp 5 . . 3
128, 9, 113imtr4i 200 . 2
1312ssriv 3101 1
 Colors of variables: wff set class Syntax hints:   wb 104   wcel 1480  wral 2416  wrex 2417  cvv 2686   wss 3071  ciun 3813  ciin 3814 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-iun 3815  df-iin 3816 This theorem is referenced by: (None)
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