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Theorem iunin1 3967
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3955 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
iunin1
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem iunin1
StepHypRef Expression
1 iunin2 3966 . 2
2 incom 3361 . . . 4
32a1i 10 . . 3
43iuneq2i 3923 . 2
5 incom 3361 . 2
61, 4, 53eqtr4i 2313 1
 Colors of variables: wff set class Syntax hints:   wceq 1623   wcel 1684   cin 3151  ciun 3905 This theorem is referenced by:  2iunin  3970  tgrest  16890  metnrmlem3  18365  limciun  19244  measinblem  23547  sstotbnd2  26498 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-iun 3907
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