Definition df-inf 6983

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 6981	. 2 class inf ( A , B , R )
5	3	ccnv 4625	. . 3 class `' R
6	1, 2, 5	csup 6980	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1353	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  7009  infeq2  7012  infeq3  7013  infeq123d  7014  nfinf  7015  eqinfti  7018  infvalti  7020  infclti  7021  inflbti  7022  infglbti  7023  infsnti  7028  inf00  7029  infisoti  7030  infex2g  7032  dfinfre  8912  infrenegsupex  9593  infxrnegsupex  11270