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Theorem inf00 7024
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 6978 . 2  |- inf ( B ,  (/) ,  R )  =  sup ( B ,  (/) ,  `' R
)
2 sup00 6996 . 2  |-  sup ( B ,  (/) ,  `' R )  =  (/)
31, 2eqtri 2198 1  |- inf ( B ,  (/) ,  R )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1353   (/)c0 3422   `'ccnv 4622   supcsup 6975  infcinf 6976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-uni 3808  df-sup 6977  df-inf 6978
This theorem is referenced by: (None)
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