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Theorem xpriindi 4987
 Description: Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
xpriindi
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem xpriindi
StepHypRef Expression
1 iineq1 4311 . . . . . . 7
2 0iin 4354 . . . . . . 7
31, 2syl6eq 2479 . . . . . 6
43ineq2d 3664 . . . . 5
5 inv1 3789 . . . . 5
64, 5syl6eq 2479 . . . 4
76xpeq2d 4874 . . 3
8 iineq1 4311 . . . . . 6
9 0iin 4354 . . . . . 6
108, 9syl6eq 2479 . . . . 5
1110ineq2d 3664 . . . 4
12 inv1 3789 . . . 4
1311, 12syl6eq 2479 . . 3
147, 13eqtr4d 2466 . 2
15 xpindi 4984 . . 3
16 xpiindi 4986 . . . 4
1716ineq2d 3664 . . 3
1815, 17syl5eq 2475 . 2
1914, 18pm2.61ine 2737 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1437   wne 2618  cvv 3081   cin 3435  c0 3761  ciin 4297   cxp 4848 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-iin 4299  df-opab 4480  df-xp 4856  df-rel 4857 This theorem is referenced by: (None)
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