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Theorem indir 3180
Description: Distributive law for intersection over union. Theorem 28 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
indir  u.  i^i  C  i^i  C  u.  i^i  C

Proof of Theorem indir
StepHypRef Expression
1 indi 3178 . 2  C  i^i  u.  C  i^i  u.  C  i^i
2 incom 3123 . 2  u.  i^i  C  C  i^i  u.
3 incom 3123 . . 3  i^i  C  C  i^i
4 incom 3123 . . 3  i^i  C  C  i^i
53, 4uneq12i 3089 . 2  i^i  C  u.  i^i  C  C  i^i  u.  C  i^i
61, 2, 53eqtr4i 2067 1  u.  i^i  C  i^i  C  u.  i^i  C
Colors of variables: wff set class
Syntax hints:   wceq 1242    u. cun 2909    i^i cin 2910
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-bndl 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-v 2553  df-un 2916  df-in 2918
This theorem is referenced by:  difundir  3184  undisj1  3273  disjpr2  3425  resundir  4569
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