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Theorem uniin 3563
Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniin  U.  i^i  C_  U.  i^i  U.

Proof of Theorem uniin
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.40 1495 . . . 4
2 elin 3094 . . . . . . 7  i^i
32anbi2i 430 . . . . . 6  i^i
4 anandi 509 . . . . . 6
53, 4bitri 173 . . . . 5  i^i
65exbii 1469 . . . 4  i^i
7 eluni 3546 . . . . 5  U.
8 eluni 3546 . . . . 5  U.
97, 8anbi12i 433 . . . 4  U.  U.
101, 6, 93imtr4i 190 . . 3  i^i  U.  U.
11 eluni 3546 . . 3  U.  i^i  i^i
12 elin 3094 . . 3  U.  i^i  U.  U.  U.
1310, 11, 123imtr4i 190 . 2  U.  i^i  U.  i^i  U.
1413ssriv 2917 1  U.  i^i  C_  U.  i^i  U.
Colors of variables: wff set class
Syntax hints:   wa 97  wex 1354   wcel 1366    i^i cin 2884    C_ wss 2885   U.cuni 3543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-in 2892  df-ss 2899  df-uni 3544
This theorem is referenced by: (None)
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