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Theorem xpundir 5058
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 5010 . 2
2 df-xp 5010 . . . 4
3 df-xp 5010 . . . 4
42, 3uneq12i 3655 . . 3
5 elun 3644 . . . . . . 7
65anbi1i 695 . . . . . 6
7 andir 868 . . . . . 6
86, 7bitri 249 . . . . 5
98opabbii 4516 . . . 4
10 unopab 4527 . . . 4
119, 10eqtr4i 2489 . . 3
124, 11eqtr4i 2489 . 2
131, 12eqtr4i 2489 1
Colors of variables: wff setvar class
Syntax hints:  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  u.cun 3473  {copab 4509  X.cxp 5002
This theorem is referenced by:  xpun  5062  resundi  5292  xpfi  7811  cdaassen  8583  hashxplem  12491  ustund  20724  cnmpt2pc  21428  pwssplit4  31035  xpprsng  32921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-opab 4511  df-xp 5010
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