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Theorem xp0 5018
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 4679 . . 3 (∅ × 𝐴) = ∅
21cnveqi 4774 . 2 (∅ × 𝐴) =
3 cnvxp 5017 . 2 (∅ × 𝐴) = (𝐴 × ∅)
4 cnv0 5002 . 2 ∅ = ∅
52, 3, 43eqtr3i 2193 1 (𝐴 × ∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1342  c0 3405   × cxp 4597  ccnv 4598
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-xp 4605  df-rel 4606  df-cnv 4607
This theorem is referenced by:  xpeq0r  5021  xpdisj2  5024  djuassen  7165  xpdjuen  7166  0met  12951
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