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Theorem xpundir 4397
 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 4351 . 2
2 df-xp 4351 . . . 4
3 df-xp 4351 . . . 4
42, 3uneq12i 3095 . . 3
5 elun 3084 . . . . . . 7
65anbi1i 431 . . . . . 6
7 andir 732 . . . . . 6
86, 7bitri 173 . . . . 5
98opabbii 3824 . . . 4
10 unopab 3836 . . . 4
119, 10eqtr4i 2063 . . 3
124, 11eqtr4i 2063 . 2
131, 12eqtr4i 2063 1
 Colors of variables: wff set class Syntax hints:   wa 97   wo 629   wceq 1243   wcel 1393   cun 2915  copab 3817   cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-opab 3819  df-xp 4351 This theorem is referenced by:  xpun  4401  resundi  4625
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