Theorem inf00 9454

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 9389	. 2 ⊢ inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, ◡𝑅)
2		sup00 9411	. 2 ⊢ sup(𝐵, ∅, ◡𝑅) = ∅
3	1, 2	eqtri 2785	1 ⊢ inf(𝐵, ∅, 𝑅) = ∅

Syntax hints:   = wceq 1560  ∅c0 4285  ^◡ccnv 5646  supcsup 9386  infcinf 9387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-ss 3921  df-nul 4286  df-uni 4866  df-sup 9388  df-inf 9389

This theorem is referenced by: (None)