ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inf00 Unicode version

Theorem inf00 6924
Description: The infimum regarding an empty base 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 6878 . 2  |- inf ( B ,  (/) ,  R )  =  sup ( B ,  (/) ,  `' R
)
2 sup00 6896 . 2  |-  sup ( B ,  (/) ,  `' R )  =  (/)
31, 2eqtri 2161 1  |- inf ( B ,  (/) ,  R )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1332   (/)c0 3366   `'ccnv 4544   supcsup 6875  infcinf 6876
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3076  df-in 3080  df-ss 3087  df-nul 3367  df-sn 3536  df-uni 3743  df-sup 6877  df-inf 6878
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator