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Definition df-trans 5899
 Description: Define the set of all transitive relationships over a base set. (Contributed by SF, 19-Feb-2015.)
Assertion
Ref Expression
df-trans Trans = {r, a x a y a z a ((xry yrz) → xrz)}
Distinct variable group:   r,a,x,y,z

Detailed syntax breakdown of Definition df-trans
StepHypRef Expression
1 ctrans 5888 . 2 class Trans
2 vx . . . . . . . . . 10 setvar x
32cv 1641 . . . . . . . . 9 class x
4 vy . . . . . . . . . 10 setvar y
54cv 1641 . . . . . . . . 9 class y
6 vr . . . . . . . . . 10 setvar r
76cv 1641 . . . . . . . . 9 class r
83, 5, 7wbr 4639 . . . . . . . 8 wff xry
9 vz . . . . . . . . . 10 setvar z
109cv 1641 . . . . . . . . 9 class z
115, 10, 7wbr 4639 . . . . . . . 8 wff yrz
128, 11wa 358 . . . . . . 7 wff (xry yrz)
133, 10, 7wbr 4639 . . . . . . 7 wff xrz
1412, 13wi 4 . . . . . 6 wff ((xry yrz) → xrz)
15 va . . . . . . 7 setvar a
1615cv 1641 . . . . . 6 class a
1714, 9, 16wral 2614 . . . . 5 wff z a ((xry yrz) → xrz)
1817, 4, 16wral 2614 . . . 4 wff y a z a ((xry yrz) → xrz)
1918, 2, 16wral 2614 . . 3 wff x a y a z a ((xry yrz) → xrz)
2019, 6, 15copab 4622 . 2 class {r, a x a y a z a ((xry yrz) → xrz)}
211, 20wceq 1642 1 wff Trans = {r, a x a y a z a ((xry yrz) → xrz)}
 Colors of variables: wff setvar class This definition is referenced by:  transex  5910  trd  5921  trrd  5925
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