ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xp0 GIF version

Theorem xp0 4793
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 4466 . . 3 (∅ × 𝐴) = ∅
21cnveqi 4558 . 2 (∅ × 𝐴) =
3 cnvxp 4792 . 2 (∅ × 𝐴) = (𝐴 × ∅)
4 cnv0 4777 . 2 ∅ = ∅
52, 3, 43eqtr3i 2111 1 (𝐴 × ∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1285  c0 3267   × cxp 4389  ccnv 4390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399
This theorem is referenced by:  xpeq0r  4796  xpdisj2  4798
  Copyright terms: Public domain W3C validator