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

Theorem inuni 4556
 Description: The intersection of a union with a class is equal to the union of the intersections of each element of with . (Contributed by FL, 24-Mar-2007.)
Assertion
Ref Expression
inuni
Distinct variable groups:   ,,   ,,

Proof of Theorem inuni
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eluni2 4195 . . . . 5
21anbi1i 693 . . . 4
3 elin 3626 . . . 4
4 ancom 448 . . . . . . . 8
5 r19.41v 2959 . . . . . . . 8
64, 5bitr4i 252 . . . . . . 7
76exbii 1688 . . . . . 6
8 rexcom4 3079 . . . . . 6
97, 8bitr4i 252 . . . . 5
10 vex 3062 . . . . . . . . . 10
1110inex1 4535 . . . . . . . . 9
12 eleq2 2475 . . . . . . . . 9
1311, 12ceqsexv 3096 . . . . . . . 8
14 elin 3626 . . . . . . . 8
1513, 14bitri 249 . . . . . . 7
1615rexbii 2906 . . . . . 6
17 r19.41v 2959 . . . . . 6
1816, 17bitri 249 . . . . 5
199, 18bitri 249 . . . 4
202, 3, 193bitr4i 277 . . 3
21 eluniab 4202 . . 3
2220, 21bitr4i 252 . 2
2322eqriv 2398 1
 Colors of variables: wff setvar class Syntax hints:   wa 367   wceq 1405  wex 1633   wcel 1842  cab 2387  wrex 2755   cin 3413  cuni 4191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-v 3061  df-in 3421  df-uni 4192 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator