Description: Trichotomy law. (Contributed by NM, 18-Aug-1999) (Proof shortened by Andrew Salmon, 19-Nov-2011)
Ref | Expression | ||
---|---|---|---|
Assertion | letric | |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B \/ B <_ A ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnle | |- ( ( B e. RR /\ A e. RR ) -> ( B < A <-> -. A <_ B ) ) |
|
2 | ltle | |- ( ( B e. RR /\ A e. RR ) -> ( B < A -> B <_ A ) ) |
|
3 | 1 2 | sylbird | |- ( ( B e. RR /\ A e. RR ) -> ( -. A <_ B -> B <_ A ) ) |
4 | 3 | orrd | |- ( ( B e. RR /\ A e. RR ) -> ( A <_ B \/ B <_ A ) ) |
5 | 4 | ancoms | |- ( ( A e. RR /\ B e. RR ) -> ( A <_ B \/ B <_ A ) ) |