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Theorem uniin 4269
 Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 7410 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniin

Proof of Theorem uniin
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1679 . . . 4
2 elin 3686 . . . . . . 7
32anbi2i 694 . . . . . 6
4 anandi 828 . . . . . 6
53, 4bitri 249 . . . . 5
65exbii 1667 . . . 4
7 eluni 4252 . . . . 5
8 eluni 4252 . . . . 5
97, 8anbi12i 697 . . . 4
101, 6, 93imtr4i 266 . . 3
11 eluni 4252 . . 3
12 elin 3686 . . 3
1310, 11, 123imtr4i 266 . 2
1413ssriv 3507 1
 Colors of variables: wff setvar class Syntax hints:  /\wa 369  E.wex 1612  e.wcel 1818  i^icin 3474  C_wss 3475  U.cuni 4249 This theorem is referenced by:  uniinqs  7410  psss  15844  tgval  19456  mapdunirnN  37377 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-in 3482  df-ss 3489  df-uni 4250
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