Definition df-inf 7276

Description: Define the infimum of class A. It is meaningful when R is a relation that strictly orders B and when the infimum exists. For example, R could be 'less than', B could be the set of real numbers, and A could be the set of all positive reals; in this case the infimum is 0. The infimum is defined as the supremum using the converse ordering relation. In the given example, 0 is the supremum of all reals (greatest real number) for which all positive reals are greater. (Contributed by AV, 2-Sep-2020.)

Assertion

Ref	Expression
df-inf	|- inf ( A , B , R ) = sup ( A , B , `' R )

Detailed syntax breakdown of Definition df-inf

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3		cR	. . 3 class R
4	1, 2, 3	cinf 7274	. 2 class inf ( A , B , R )
5	3	ccnv 4748	. . 3 class `' R
6	1, 2, 5	csup 7273	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1398	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  7302  infeq2  7305  infeq3  7306  infeq123d  7307  nfinf  7308  eqinfti  7311  infvalti  7313  infclti  7314  inflbti  7315  infglbti  7316  infsnti  7321  inf00  7322  infisoti  7323  infex2g  7325  dfinfre  9230  infrenegsupex  9926  infxrnegsupex  11948