Definition df-inf 7163

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 7161	. 2 class inf ( A , B , R )
5	3	ccnv 4718	. . 3 class `' R
6	1, 2, 5	csup 7160	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1395	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  7189  infeq2  7192  infeq3  7193  infeq123d  7194  nfinf  7195  eqinfti  7198  infvalti  7200  infclti  7201  inflbti  7202  infglbti  7203  infsnti  7208  inf00  7209  infisoti  7210  infex2g  7212  dfinfre  9114  infrenegsupex  9801  infxrnegsupex  11789