Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  inf00 Structured version   Visualization version   GIF version

Theorem inf00 8294
 Description: The infimum regarding an empty bas set is always the empty set. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
inf00 inf(𝐵, ∅, 𝑅) = ∅

Proof of Theorem inf00
StepHypRef Expression
1 df-inf 8232 . 2 inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, 𝑅)
2 sup00 8253 . 2 sup(𝐵, ∅, 𝑅) = ∅
31, 2eqtri 2632 1 inf(𝐵, ∅, 𝑅) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∅c0 3874  ◡ccnv 5037  supcsup 8229  infcinf 8230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-uni 4373  df-sup 8231  df-inf 8232 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator