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Theorem uniin 3863
 Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uninqs 25142 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 1599 . . . 4
2 elin 3371 . . . . . . 7
32anbi2i 675 . . . . . 6
4 anandi 801 . . . . . 6
53, 4bitri 240 . . . . 5
65exbii 1572 . . . 4
7 eluni 3846 . . . . 5
8 eluni 3846 . . . . 5
97, 8anbi12i 678 . . . 4
101, 6, 93imtr4i 257 . . 3
11 eluni 3846 . . 3
12 elin 3371 . . 3
1310, 11, 123imtr4i 257 . 2
1413ssriv 3197 1
 Colors of variables: wff set class Syntax hints:   wa 358  wex 1531   wcel 1696   cin 3164   wss 3165  cuni 3843 This theorem is referenced by:  psss  14339  tgval  16709  uninqs  25142  uuniin  25190  inposet  25381  unint2t  25621  mapdunirnN  32462 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-uni 3844
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