Definition df-inf 6946

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 6944	. 2 class inf ( A , B , R )
5	3	ccnv 4602	. . 3 class `' R
6	1, 2, 5	csup 6943	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1343	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  6972  infeq2  6975  infeq3  6976  infeq123d  6977  nfinf  6978  eqinfti  6981  infvalti  6983  infclti  6984  inflbti  6985  infglbti  6986  infsnti  6991  inf00  6992  infisoti  6993  infex2g  6995  dfinfre  8847  infrenegsupex  9528  infxrnegsupex  11200