Definition df-inf 7148

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 7146	. 2 class inf ( A , B , R )
5	3	ccnv 4717	. . 3 class `' R
6	1, 2, 5	csup 7145	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1395	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  7174  infeq2  7177  infeq3  7178  infeq123d  7179  nfinf  7180  eqinfti  7183  infvalti  7185  infclti  7186  inflbti  7187  infglbti  7188  infsnti  7193  inf00  7194  infisoti  7195  infex2g  7197  dfinfre  9099  infrenegsupex  9785  infxrnegsupex  11769