Definition df-inf 7102

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 7100	. 2 class inf ( A , B , R )
5	3	ccnv 4682	. . 3 class `' R
6	1, 2, 5	csup 7099	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1373	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  7128  infeq2  7131  infeq3  7132  infeq123d  7133  nfinf  7134  eqinfti  7137  infvalti  7139  infclti  7140  inflbti  7141  infglbti  7142  infsnti  7147  inf00  7148  infisoti  7149  infex2g  7151  dfinfre  9049  infrenegsupex  9735  infxrnegsupex  11649