Metamath Proof Explorer


Theorem eqleltd

Description: Equality in terms of 'less than or equal to', 'less than'. (Contributed by NM, 7-Apr-2001)

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

Proof

Step Hyp Ref Expression
1 ltd.1
 |-  ( ph -> A e. RR )
2 ltd.2
 |-  ( ph -> B e. RR )
3 eqlelt
 |-  ( ( A e. RR /\ B e. RR ) -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( A = B <-> ( A <_ B /\ -. A < B ) ) )