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Theorem inf00 7023
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 6977 . 2 inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, 𝑅)
2 sup00 6995 . 2 sup(𝐵, ∅, 𝑅) = ∅
31, 2eqtri 2198 1 inf(𝐵, ∅, 𝑅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1353  c0 3422  ccnv 4621  supcsup 6974  infcinf 6975
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 6976  df-inf 6977
This theorem is referenced by: (None)
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