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Theorem iunin1 3712
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 3701 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
iunin1  U_  C  i^i  U_  C  i^i
Distinct variable group:   ,
Allowed substitution hints:   ()    C()

Proof of Theorem iunin1
StepHypRef Expression
1 iunin2 3711 . 2  U_  i^i  C  i^i  U_  C
2 incom 3123 . . . 4  C  i^i  i^i  C
32a1i 9 . . 3  C  i^i  i^i  C
43iuneq2i 3666 . 2  U_  C  i^i  U_  i^i  C
5 incom 3123 . 2  U_  C  i^i  i^i  U_  C
61, 4, 53eqtr4i 2067 1  U_  C  i^i  U_  C  i^i
Colors of variables: wff set class
Syntax hints:   wceq 1242   wcel 1390    i^i cin 2910   U_ciun 3648
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-iun 3650
This theorem is referenced by:  2iunin  3714
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