Theorem inf00 9411

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

Step	Hyp	Ref	Expression
1		df-inf 9346	. 2 ⊢ inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, ◡𝑅)
2		sup00 9368	. 2 ⊢ sup(𝐵, ∅, ◡𝑅) = ∅
3	1, 2	eqtri 2762	1 ⊢ inf(𝐵, ∅, 𝑅) = ∅

Syntax hints:   = wceq 1547  ∅c0 4261  ^◡ccnv 5617  supcsup 9343  infcinf 9344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-ss 3900  df-nul 4262  df-uni 4839  df-sup 9345  df-inf 9346

This theorem is referenced by: (None)