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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 ) |