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Theorem inxp 4498
 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 4496 . . 3
2 an4 528 . . . . 5
3 elin 3154 . . . . . 6
4 elin 3154 . . . . . 6
53, 4anbi12i 441 . . . . 5
62, 5bitr4i 180 . . . 4
76opabbii 3852 . . 3
81, 7eqtri 2076 . 2
9 df-xp 4379 . . 3
10 df-xp 4379 . . 3
119, 10ineq12i 3164 . 2
12 df-xp 4379 . 2
138, 11, 123eqtr4i 2086 1
 Colors of variables: wff set class Syntax hints:   wa 101   wceq 1259   wcel 1409   cin 2944  copab 3845   cxp 4371 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-rel 4380 This theorem is referenced by:  xpindi  4499  xpindir  4500  dmxpinm  4584  xpssres  4673  xpdisj1  4775  xpdisj2  4776  imainrect  4794  xpima1  4795  xpima2m  4796
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