New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  xpundir Unicode version

Theorem xpundir 4833
 Description: Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
xpundir

Proof of Theorem xpundir
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elun 3220 . . . . . 6
21anbi1i 676 . . . . 5
3 andir 838 . . . . 5
42, 3bitri 240 . . . 4
54opabbii 4626 . . 3
6 unopab 4638 . . 3
75, 6eqtr4i 2376 . 2
8 df-xp 4784 . 2
9 df-xp 4784 . . 3
10 df-xp 4784 . . 3
119, 10uneq12i 3416 . 2
127, 8, 113eqtr4i 2383 1
 Colors of variables: wff setvar class Syntax hints:   wo 357   wa 358   wceq 1642   wcel 1710   cun 3207  copab 4622   cxp 4770 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-un 3214  df-opab 4623  df-xp 4784 This theorem is referenced by:  xpun  4834  resundi  4981
 Copyright terms: Public domain W3C validator