Theorem inf00 7230

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 7184	. 2 |- inf ( B , (/) , R ) = sup ( B , (/) , `' R )
2		sup00 7202	. 2 |- sup ( B , (/) , `' R ) = (/)
3	1, 2	eqtri 2252	1 |- inf ( B , (/) , R ) = (/)

Syntax hints:   = wceq 1397   (/)c0 3494   `'ccnv 4724   supcsup 7181  infcinf 7182

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-sn 3675  df-uni 3894  df-sup 7183  df-inf 7184

This theorem is referenced by: (None)