Metamath Proof Explorer


Theorem suble0d

Description: Nonpositive subtraction. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1
|- ( ph -> A e. RR )
ltnegd.2
|- ( ph -> B e. RR )
Assertion suble0d
|- ( ph -> ( ( A - B ) <_ 0 <-> A <_ B ) )

Proof

Step Hyp Ref Expression
1 leidd.1
 |-  ( ph -> A e. RR )
2 ltnegd.2
 |-  ( ph -> B e. RR )
3 suble0
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( A - B ) <_ 0 <-> A <_ B ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( ( A - B ) <_ 0 <-> A <_ B ) )