Theorem inf00 7322

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 7276	. 2 |- inf ( B , (/) , R ) = sup ( B , (/) , `' R )
2		sup00 7294	. 2 |- sup ( B , (/) , `' R ) = (/)
3	1, 2	eqtri 2253	1 |- inf ( B , (/) , R ) = (/)

Syntax hints:   = wceq 1398   (/)c0 3508   `'ccnv 4748   supcsup 7273  infcinf 7274

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-nul 3509  df-sn 3695  df-uni 3915  df-sup 7275  df-inf 7276

This theorem is referenced by: (None)