Definition df-inf 7331

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 7329	. 2 class inf ( A , B , R )
5	3	ccnv 4775	. . 3 class `' R
6	1, 2, 5	csup 7328	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1402	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is used by:  infeq1  7357  infeq2  7360  infeq3  7361  infeq123d  7362  nfinf  7363  eqinfti  7366  infvalti  7368  infclti  7369  inflbti  7370  infglbti  7371  infsnti  7376  inf00  7377  infisoti  7378  infex2g  7380  dfinfre  9294  infrenegsupex  10010  infxrnegsupex  12054