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Theorem xp0 4953
 Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
Assertion
Ref Expression
xp0 (𝐴 × ∅) = ∅

Proof of Theorem xp0
StepHypRef Expression
1 0xp 4614 . . 3 (∅ × 𝐴) = ∅
21cnveqi 4709 . 2 (∅ × 𝐴) =
3 cnvxp 4952 . 2 (∅ × 𝐴) = (𝐴 × ∅)
4 cnv0 4937 . 2 ∅ = ∅
52, 3, 43eqtr3i 2166 1 (𝐴 × ∅) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1331  ∅c0 3358   × cxp 4532  ◡ccnv 4533 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-in1 603  ax-in2 604  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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542 This theorem is referenced by:  xpeq0r  4956  xpdisj2  4959  djuassen  7066  xpdjuen  7067  0met  12542
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