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Theorem uniin 4239
 Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 7454 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 1725 . . . 4
2 elin 3649 . . . . . . 7
32anbi2i 698 . . . . . 6
4 anandi 835 . . . . . 6
53, 4bitri 252 . . . . 5
65exbii 1712 . . . 4
7 eluni 4222 . . . . 5
8 eluni 4222 . . . . 5
97, 8anbi12i 701 . . . 4
101, 6, 93imtr4i 269 . . 3
11 eluni 4222 . . 3
12 elin 3649 . . 3
1310, 11, 123imtr4i 269 . 2
1413ssriv 3468 1
 Colors of variables: wff setvar class Syntax hints:   wa 370  wex 1657   wcel 1872   cin 3435   wss 3436  cuni 4219 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-in 3443  df-ss 3450  df-uni 4220 This theorem is referenced by:  uniinqs  7454  psss  16459  tgval  19968  mapdunirnN  35187
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