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Theorem 0nelelxp 4401
 Description: A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.)
Assertion
Ref Expression
0nelelxp

Proof of Theorem 0nelelxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4390 . 2
2 0nelop 4013 . . . 4
3 simpl 106 . . . . 5
43eleq2d 2123 . . . 4
52, 4mtbiri 610 . . 3
65exlimivv 1792 . 2
71, 6sylbi 118 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 101   wceq 1259  wex 1397   wcel 1409  c0 3252  cop 3406   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-in1 554  ax-in2 555  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-ne 2221  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379 This theorem is referenced by:  dmsn0el  4818
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