Definition df-irreflexive 42511

Description: Define irreflexive relation; relation R is irreflexive over the set A iff A. x e. A -. x R x. Note that a relation can be neither reflexive nor irreflexive. (Contributed by David A. Wheeler, 1-Dec-2019.)

Assertion

Ref	Expression
df-irreflexive	|- ( RIrreflexive A <-> ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) )

Distinct variable groups:   x, A   x, R

Detailed syntax breakdown of Definition df-irreflexive

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cR	. . 3 class R
3	1, 2	wirreflexive 42510	. 2 wff RIrreflexive A
4	1, 1	cxp 5112	. . . 4 class ( A X. A )
5	2, 4	wss 3574	. . 3 wff R C_ ( A X. A )
6		vx	. . . . . . 7 setvar x
7	6	cv 1482	. . . . . 6 class x
8	7, 7, 2	wbr 4653	. . . . 5 wff x R x
9	8	wn 3	. . . 4 wff -. x R x
10	9, 6, 1	wral 2912	. . 3 wff A. x e. A -. x R x
11	5, 10	wa 384	. 2 wff ( R C_ ( A X. A ) /\ A. x e. A -. x R x )
12	3, 11	wb 196	1 wff ( RIrreflexive A <-> ( R C_ ( A X. A ) /\ A. x e. A -. x R x ) )

This definition is referenced by: (None)