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Theorem xpundir 4917
 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 df-xp 4870 . 2
2 df-xp 4870 . . . 4
3 df-xp 4870 . . . 4
42, 3uneq12i 3486 . . 3
5 elun 3475 . . . . . . 7
65anbi1i 677 . . . . . 6
7 andir 839 . . . . . 6
86, 7bitri 241 . . . . 5
98opabbii 4259 . . . 4
10 unopab 4271 . . . 4
119, 10eqtr4i 2453 . . 3
124, 11eqtr4i 2453 . 2
131, 12eqtr4i 2453 1
 Colors of variables: wff set class Syntax hints:   wo 358   wa 359   wceq 1652   wcel 1725   cun 3305  copab 4252   cxp 4862 This theorem is referenced by:  xpun  4921  resundi  5146  xpfi  7364  cdaassen  8046  hashxplem  11679  ustund  18234  cnmpt2pc  18936  pwssplit4  27101 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 2411 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 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-v 2945  df-un 3312  df-opab 4254  df-xp 4870
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