Definition df-inf 6872

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 6870	. 2 class inf ( A , B , R )
5	3	ccnv 4538	. . 3 class `' R
6	1, 2, 5	csup 6869	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1331	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  6898  infeq2  6901  infeq3  6902  infeq123d  6903  nfinf  6904  eqinfti  6907  infvalti  6909  infclti  6910  inflbti  6911  infglbti  6912  infsnti  6917  inf00  6918  infisoti  6919  dfinfre  8714  infrenegsupex  9389  infxrnegsupex  11032