ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  unixp0im GIF version

Theorem unixp0im 5140
Description: The union of an empty cross product is empty. (Contributed by Jim Kingdon, 18-Dec-2018.)
Assertion
Ref Expression
unixp0im ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)

Proof of Theorem unixp0im
StepHypRef Expression
1 unieq 3798 . 2 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
2 uni0 3816 . 2 ∅ = ∅
31, 2eqtrdi 2215 1 ((𝐴 × 𝐵) = ∅ → (𝐴 × 𝐵) = ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1343  c0 3409   cuni 3789   × cxp 4602
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-uni 3790
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator