Definition df-inf 6981

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 6979	. 2 class inf ( A , B , R )
5	3	ccnv 4624	. . 3 class `' R
6	1, 2, 5	csup 6978	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1353	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  7007  infeq2  7010  infeq3  7011  infeq123d  7012  nfinf  7013  eqinfti  7016  infvalti  7018  infclti  7019  inflbti  7020  infglbti  7021  infsnti  7026  inf00  7027  infisoti  7028  infex2g  7030  dfinfre  8909  infrenegsupex  9590  infxrnegsupex  11264