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Theorem uniin 3747
 Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uninqs 24204 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
StepHypRef Expression
1 19.40 1608 . . . 4
2 elin 3266 . . . . . . 7
32anbi2i 678 . . . . . 6
4 anandi 804 . . . . . 6
53, 4bitri 242 . . . . 5
65exbii 1580 . . . 4
7 eluni 3730 . . . . 5
8 eluni 3730 . . . . 5
97, 8anbi12i 681 . . . 4
101, 6, 93imtr4i 259 . . 3
11 eluni 3730 . . 3
12 elin 3266 . . 3
1310, 11, 123imtr4i 259 . 2
1413ssriv 3105 1
 Colors of variables: wff set class Syntax hints:   wa 360  wex 1537   wcel 1621   cin 3077   wss 3078  cuni 3727 This theorem is referenced by:  psss  14158  tgval  16525  uninqs  24204  uuniin  24252  inposet  24444  unint2t  24684  mapdunirnN  30529 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-in 3085  df-ss 3089  df-uni 3728
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