Definition df-inf 6921

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 6919	. 2 class inf ( A , B , R )
5	3	ccnv 4582	. . 3 class `' R
6	1, 2, 5	csup 6918	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1335	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  6947  infeq2  6950  infeq3  6951  infeq123d  6952  nfinf  6953  eqinfti  6956  infvalti  6958  infclti  6959  inflbti  6960  infglbti  6961  infsnti  6966  inf00  6967  infisoti  6968  dfinfre  8810  infrenegsupex  9488  infxrnegsupex  11142