Definition df-inf 7183

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 7181	. 2 class inf ( A , B , R )
5	3	ccnv 4724	. . 3 class `' R
6	1, 2, 5	csup 7180	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1397	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  7209  infeq2  7212  infeq3  7213  infeq123d  7214  nfinf  7215  eqinfti  7218  infvalti  7220  infclti  7221  inflbti  7222  infglbti  7223  infsnti  7228  inf00  7229  infisoti  7230  infex2g  7232  dfinfre  9135  infrenegsupex  9827  infxrnegsupex  11823