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Theorem inf00 6670
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 6624 . 2  |- inf ( B ,  (/) ,  R )  =  sup ( B ,  (/) ,  `' R
)
2 sup00 6642 . 2  |-  sup ( B ,  (/) ,  `' R )  =  (/)
31, 2eqtri 2105 1  |- inf ( B ,  (/) ,  R )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1287   (/)c0 3275   `'ccnv 4410   supcsup 6621  infcinf 6622
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-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-dif 2990  df-in 2994  df-ss 3001  df-nul 3276  df-sn 3437  df-uni 3637  df-sup 6623  df-inf 6624
This theorem is referenced by: (None)
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