Theorem inf00 6987

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 6941	. 2 |- inf ( B , (/) , R ) = sup ( B , (/) , `' R )
2		sup00 6959	. 2 |- sup ( B , (/) , `' R ) = (/)
3	1, 2	eqtri 2185	1 |- inf ( B , (/) , R ) = (/)

Syntax hints:   = wceq 1342   (/)c0 3404   `'ccnv 4597   supcsup 6938  infcinf 6939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-dif 3113  df-in 3117  df-ss 3124  df-nul 3405  df-sn 3576  df-uni 3784  df-sup 6940  df-inf 6941

This theorem is referenced by: (None)