Theorem inf00 9043

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 8980	. 2 ⊢ inf(𝐵, ∅, 𝑅) = sup(𝐵, ∅, ◡𝑅)
2		sup00 9001	. 2 ⊢ sup(𝐵, ∅, ◡𝑅) = ∅
3	1, 2	eqtri 2761	1 ⊢ inf(𝐵, ∅, 𝑅) = ∅

Syntax hints:   = wceq 1542  ∅c0 4211  ^◡ccnv 5524  supcsup 8977  infcinf 8978

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 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-in 3850  df-ss 3860  df-nul 4212  df-sn 4517  df-uni 4797  df-sup 8979  df-inf 8980

This theorem is referenced by: (None)