Definition df-inf 7315

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 7313	. 2 class inf ( A , B , R )
5	3	ccnv 4768	. . 3 class `' R
6	1, 2, 5	csup 7312	. 2 class sup ( A , B , `' R )
7	4, 6	wceq 1402	1 wff inf ( A , B , R ) = sup ( A , B , `' R )

This definition is referenced by:  infeq1  7341  infeq2  7344  infeq3  7345  infeq123d  7346  nfinf  7347  eqinfti  7350  infvalti  7352  infclti  7353  inflbti  7354  infglbti  7355  infsnti  7360  inf00  7361  infisoti  7362  infex2g  7364  dfinfre  9276  infrenegsupex  9973  infxrnegsupex  12007