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Theorem xpundi 4316
Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52.
Assertion
Ref Expression
xpundi

Proof of Theorem xpundi
StepHypRef Expression
1 elun 2953 . . . . . 6
21anbi2i 671 . . . . 5
3 andi 801 . . . . 5
42, 3bitri 238 . . . 4
54opabbii 3649 . . 3
6 unopab 3658 . . 3
75, 6eqtr4i 2113 . 2
8 df-xp 4263 . 2
9 df-xp 4263 . . 3
10 df-xp 4263 . . 3
119, 10uneq12i 2964 . 2
127, 8, 113eqtr4i 2120 1
Colors of variables: wff set class
Syntax hints:   wo 356   wa 357   wceq 1531   wcel 1533   cun 2808  copab 3644   cxp 4247
This theorem is referenced by:  xpun  4322  xp2cda  7175  xpcdaen  7178  alephadd  7575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-5 1446  ax-6 1447  ax-7 1448  ax-gen 1449  ax-8 1535  ax-11 1536  ax-12 1537  ax-17 1542  ax-9 1563  ax-10 1591  ax-4 1605  ax-16 1790  ax-ext 2072
This theorem depends on definitions:  df-bi 175  df-or 358  df-an 359  df-ex 1451  df-sb 1751  df-clab 2078  df-cleq 2083  df-clel 2086  df-v 2502  df-un 2815  df-opab 3645  df-xp 4263
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