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Theorem xpundir 4649
 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
StepHypRef Expression
1 df-xp 4594 . 2
2 df-xp 4594 . . . 4
3 df-xp 4594 . . . 4
42, 3uneq12i 3237 . . 3
5 elun 3226 . . . . . . 7
65anbi1i 679 . . . . . 6
7 andir 843 . . . . . 6
86, 7bitri 242 . . . . 5
98opabbii 3980 . . . 4
10 unopab 3992 . . . 4
119, 10eqtr4i 2276 . . 3
124, 11eqtr4i 2276 . 2
131, 12eqtr4i 2276 1
 Colors of variables: wff set class Syntax hints:   wo 359   wa 360   wceq 1619   wcel 1621   cun 3076  copab 3973   cxp 4578 This theorem is referenced by:  xpun  4654  resundi  4876  xpfi  7013  cdaassen  7692  hashxplem  11262  cnmpt2pc  18258  pwssplit4  26357 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-un 3083  df-opab 3975  df-xp 4594
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