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Theorem xpundir 4903
 Description: Distributive law for Cartesian 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 df-xp 4855 . 2
2 df-xp 4855 . . . 4
3 df-xp 4855 . . . 4
42, 3uneq12i 3618 . . 3
5 elun 3606 . . . . . . 7
65anbi1i 699 . . . . . 6
7 andir 876 . . . . . 6
86, 7bitri 252 . . . . 5
98opabbii 4485 . . . 4
10 unopab 4496 . . . 4
119, 10eqtr4i 2454 . . 3
124, 11eqtr4i 2454 . 2
131, 12eqtr4i 2454 1
 Colors of variables: wff setvar class Syntax hints:   wo 369   wa 370   wceq 1437   wcel 1868   cun 3434  copab 4478   cxp 4847 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-un 3441  df-opab 4480  df-xp 4855 This theorem is referenced by:  xpun  4907  resundi  5133  xpfi  7844  cdaassen  8612  hashxplem  12602  ustund  21222  cnmpt2pc  21942  poimirlem3  31856  poimirlem4  31857  poimirlem6  31859  poimirlem7  31860  poimirlem16  31869  poimirlem19  31872  pwssplit4  35866  xpprsng  39386
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