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Theorem iunin1 3884
 Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3873 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 3883 . 2
2 incom 3272 . . . 4
32a1i 9 . . 3
43iuneq2i 3838 . 2
5 incom 3272 . 2
61, 4, 53eqtr4i 2171 1
 Colors of variables: wff set class Syntax hints:   wceq 1332   wcel 1481   cin 3074  ciun 3820 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3081  df-ss 3088  df-iun 3822 This theorem is referenced by:  2iunin  3886  tgrest  12375
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