Metamath Proof Explorer
Description: An irreflexive, transitive, linear relation is a strict ordering.
(Contributed by NM, 21-Jan-1996) (Revised by Mario Carneiro, 9-Jul-2014)
|
|
Ref |
Expression |
|
Hypotheses |
issod.1 |
|- ( ph -> R Po A ) |
|
|
issod.2 |
|- ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x R y \/ x = y \/ y R x ) ) |
|
Assertion |
issod |
|- ( ph -> R Or A ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
issod.1 |
|- ( ph -> R Po A ) |
2 |
|
issod.2 |
|- ( ( ph /\ ( x e. A /\ y e. A ) ) -> ( x R y \/ x = y \/ y R x ) ) |
3 |
2
|
ralrimivva |
|- ( ph -> A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) |
4 |
|
df-so |
|- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A ( x R y \/ x = y \/ y R x ) ) ) |
5 |
1 3 4
|
sylanbrc |
|- ( ph -> R Or A ) |