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Theorem inxp 4673
 Description: The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
inxp

Proof of Theorem inxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inopab 4671 . . 3
2 an4 575 . . . . 5
3 elin 3259 . . . . . 6
4 elin 3259 . . . . . 6
53, 4anbi12i 455 . . . . 5
62, 5bitr4i 186 . . . 4
76opabbii 3995 . . 3
81, 7eqtri 2160 . 2
9 df-xp 4545 . . 3
10 df-xp 4545 . . 3
119, 10ineq12i 3275 . 2
12 df-xp 4545 . 2
138, 11, 123eqtr4i 2170 1
 Colors of variables: wff set class Syntax hints:   wa 103   wceq 1331   wcel 1480   cin 3070  copab 3988   cxp 4537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-rel 4546 This theorem is referenced by:  xpindi  4674  xpindir  4675  dmxpin  4761  xpssres  4854  xpdisj1  4963  xpdisj2  4964  imainrect  4984  xpima1  4985  xpima2m  4986  hashxp  10584  txbas  12441  txrest  12459  metreslem  12563
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