Theorem ne0gt0d

Description: A nonzero nonnegative number is positive. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	ltd.1	|- ( ph -> A e. RR )
	ne0gt0d.2	|- ( ph -> 0 <_ A )
	ne0gt0d.3	|- ( ph -> A =/= 0 )
Assertion	ne0gt0d	|- ( ph -> 0 < A )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	|- ( ph -> A e. RR )
2		ne0gt0d.2	|- ( ph -> 0 <_ A )
3		ne0gt0d.3	|- ( ph -> A =/= 0 )
4		ne0gt0	|- ( ( A e. RR /\ 0 <_ A ) -> ( A =/= 0 <-> 0 < A ) )
5	1 2 4	syl2anc	|- ( ph -> ( A =/= 0 <-> 0 < A ) )
6	3 5	mpbid	|- ( ph -> 0 < A )