Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniin Unicode version

Theorem uniin 4027
 Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 6975 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 1619 . . . 4
2 elin 3522 . . . . . . 7
32anbi2i 676 . . . . . 6
4 anandi 802 . . . . . 6
53, 4bitri 241 . . . . 5
65exbii 1592 . . . 4
7 eluni 4010 . . . . 5
8 eluni 4010 . . . . 5
97, 8anbi12i 679 . . . 4
101, 6, 93imtr4i 258 . . 3
11 eluni 4010 . . 3
12 elin 3522 . . 3
1310, 11, 123imtr4i 258 . 2
1413ssriv 3344 1
 Colors of variables: wff set class Syntax hints:   wa 359  wex 1550   wcel 1725   cin 3311   wss 3312  cuni 4007 This theorem is referenced by:  uniinqs  6975  psss  14634  tgval  17008  mapdunirnN  32287 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-ss 3326  df-uni 4008
 Copyright terms: Public domain W3C validator