ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpeq0r GIF version

Theorem xpeq0r 4822
Description: A cross product is empty if at least one member is empty. (Contributed by Jim Kingdon, 12-Dec-2018.)
Assertion
Ref Expression
xpeq0r ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅)

Proof of Theorem xpeq0r
StepHypRef Expression
1 xpeq1 4427 . . 3 (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
2 0xp 4488 . . 3 (∅ × 𝐵) = ∅
31, 2syl6eq 2133 . 2 (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
4 xpeq2 4428 . . 3 (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5 xp0 4819 . . 3 (𝐴 × ∅) = ∅
64, 5syl6eq 2133 . 2 (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
73, 6jaoi 669 1 ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 × 𝐵) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 662   = wceq 1287  c0 3275   × cxp 4411
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-xp 4419  df-rel 4420  df-cnv 4421
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator