Metamath Proof Explorer


Theorem letri3i

Description: Consequence of trichotomy. (Contributed by NM, 14-May-1999)

Ref Expression
Hypotheses lt.1
|- A e. RR
lt.2
|- B e. RR
Assertion letri3i
|- ( A = B <-> ( A <_ B /\ B <_ A ) )

Proof

Step Hyp Ref Expression
1 lt.1
 |-  A e. RR
2 lt.2
 |-  B e. RR
3 letri3
 |-  ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
4 1 2 3 mp2an
 |-  ( A = B <-> ( A <_ B /\ B <_ A ) )