Theorem inf00 7273

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 7227	. 2 |- inf ( B , (/) , R ) = sup ( B , (/) , `' R )
2		sup00 7245	. 2 |- sup ( B , (/) , `' R ) = (/)
3	1, 2	eqtri 2252	1 |- inf ( B , (/) , R ) = (/)

Syntax hints:   = wceq 1398   (/)c0 3496   `'ccnv 4730   supcsup 7224  infcinf 7225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-uni 3899  df-sup 7226  df-inf 7227

This theorem is referenced by: (None)