Theorem inf00 9528

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 9465	. 2 ⊢ inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, ◡𝑅)
2		sup00 9486	. 2 ⊢ sup(𝐵, ∅, ◡𝑅) = ∅
3	1, 2	eqtri 2757	1 ⊢ inf(𝐵, ∅, 𝑅) = ∅

Syntax hints:   = wceq 1539  ∅c0 4313  ^◡ccnv 5664  supcsup 9462  infcinf 9463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-ss 3948  df-nul 4314  df-sn 4607  df-uni 4888  df-sup 9464  df-inf 9465

This theorem is referenced by: (None)