Theorem inf00 6539

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 6493	. 2 |- inf ( B , (/) , R ) = sup ( B , (/) , `' R )
2		sup00 6511	. 2 |- sup ( B , (/) , `' R ) = (/)
3	1, 2	eqtri 2103	1 |- inf ( B , (/) , R ) = (/)

Syntax hints:   = wceq 1285   (/)c0 3267   `'ccnv 4390   supcsup 6490  infcinf 6491

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-in 2988  df-ss 2995  df-nul 3268  df-sn 3422  df-uni 3622  df-sup 6492  df-inf 6493

This theorem is referenced by: (None)