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Theorem inf00 9265
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 9202 . 2 inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, 𝑅)
2 sup00 9223 . 2 sup(𝐵, ∅, 𝑅) = ∅
31, 2eqtri 2766 1 inf(𝐵, ∅, 𝑅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  c0 4256  ccnv 5588  supcsup 9199  infcinf 9200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-uni 4840  df-sup 9201  df-inf 9202
This theorem is referenced by: (None)
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