Definition df-inf 7069

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 7067	. 2 class inf ( A , B , R )
5	3	ccnv 4672	. . 3 class `' R
6	1, 2, 5	csup 7066	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1372	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  7095  infeq2  7098  infeq3  7099  infeq123d  7100  nfinf  7101  eqinfti  7104  infvalti  7106  infclti  7107  inflbti  7108  infglbti  7109  infsnti  7114  inf00  7115  infisoti  7116  infex2g  7118  dfinfre  9011  infrenegsupex  9697  infxrnegsupex  11493