Theorem inf00 9459

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 9394	. 2 ⊢ inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, ◡𝑅)
2		sup00 9416	. 2 ⊢ sup(𝐵, ∅, ◡𝑅) = ∅
3	1, 2	eqtri 2752	1 ⊢ inf(𝐵, ∅, 𝑅) = ∅

Syntax hints:   = wceq 1540  ∅c0 4296  ^◡ccnv 5637  supcsup 9391  infcinf 9392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-ss 3931  df-nul 4297  df-sn 4590  df-uni 4872  df-sup 9393  df-inf 9394

This theorem is referenced by: (None)