Definition df-inf 7086

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 7084	. 2 class inf ( A , B , R )
5	3	ccnv 4673	. . 3 class `' R
6	1, 2, 5	csup 7083	. 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  7112  infeq2  7115  infeq3  7116  infeq123d  7117  nfinf  7118  eqinfti  7121  infvalti  7123  infclti  7124  inflbti  7125  infglbti  7126  infsnti  7131  inf00  7132  infisoti  7133  infex2g  7135  dfinfre  9028  infrenegsupex  9714  infxrnegsupex  11516