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Theorem uniin 3911
 Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs in set.mm 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 1609 . . . 4
2 elin 3219 . . . . . . 7
32anbi2i 675 . . . . . 6
4 anandi 801 . . . . . 6
53, 4bitri 240 . . . . 5
65exbii 1582 . . . 4
7 eluni 3894 . . . . 5
8 eluni 3894 . . . . 5
97, 8anbi12i 678 . . . 4
101, 6, 93imtr4i 257 . . 3
11 eluni 3894 . . 3
12 elin 3219 . . 3
1310, 11, 123imtr4i 257 . 2
1413ssriv 3277 1
 Colors of variables: wff setvar class Syntax hints:   wa 358  wex 1541   wcel 1710   cin 3208   wss 3257  cuni 3891 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-uni 3892 This theorem is referenced by: (None)
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