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Theorem undi 3182
Description: Distributive law for union over intersection. Exercise 11 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Sep-2002.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
undi  u.  i^i  C  u.  i^i  u.  C

Proof of Theorem undi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elin 3123 . . . 4  i^i  C  C
21orbi2i 679 . . 3  i^i  C  C
3 ordi 729 . . 3  C  C
4 elin 3123 . . . 4  u.  i^i  u.  C  u.  u.  C
5 elun 3081 . . . . 5  u.
6 elun 3081 . . . . 5  u.  C  C
75, 6anbi12i 433 . . . 4  u.  u.  C  C
84, 7bitr2i 174 . . 3  C  u.  i^i  u.  C
92, 3, 83bitri 195 . 2  i^i  C  u.  i^i  u.  C
109uneqri 3082 1  u.  i^i  C  u.  i^i  u.  C
Colors of variables: wff set class
Syntax hints:   wa 97   wo 629   wceq 1243   wcel 1393    u. cun 2912    i^i cin 2913
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919  df-in 2921
This theorem is referenced by:  undir  3184
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