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Theorem iuniin 3979
 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 2727 . . . 4
2 vex 2862 . . . . . 6
3 eliin 3974 . . . . . 6
42, 3ax-mp 8 . . . . 5
54rexbii 2639 . . . 4
6 eliun 3973 . . . . 5
76ralbii 2638 . . . 4
81, 5, 73imtr4i 257 . . 3
9 eliun 3973 . . 3
10 eliin 3974 . . . 4
112, 10ax-mp 8 . . 3
128, 9, 113imtr4i 257 . 2
1312ssriv 3277 1
 Colors of variables: wff setvar class Syntax hints:   wb 176   wcel 1710  wral 2614  wrex 2615  cvv 2859   wss 3257  ciun 3969  ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-iun 3971  df-iin 3972 This theorem is referenced by: (None)
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