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Metamath Proof Explorer

Theorem ne0gt0

Description: A nonzero nonnegative number is positive. (Contributed by NM, 20-Nov-2007)

Ref Expression
Assertion ne0gt0 
|- ( ( A e. RR /\ 0 <_ A ) -> ( A =/= 0 <-> 0 < A ) )

Proof

Step Hyp Ref Expression
1 0re 
 |-  0 e. RR
2 lttri2 
 |-  ( ( A e. RR /\ 0 e. RR ) -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
3 1 2 mpan2 
 |-  ( A e. RR -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
4 3 adantr 
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( A =/= 0 <-> ( A < 0 \/ 0 < A ) ) )
5 lenlt 
 |-  ( ( 0 e. RR /\ A e. RR ) -> ( 0 <_ A <-> -. A < 0 ) )
6 1 5 mpan 
 |-  ( A e. RR -> ( 0 <_ A <-> -. A < 0 ) )
7 6 biimpa 
 |-  ( ( A e. RR /\ 0 <_ A ) -> -. A < 0 )
8 biorf 
 |-  ( -. A < 0 -> ( 0 < A <-> ( A < 0 \/ 0 < A ) ) )
9 7 8 syl 
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( 0 < A <-> ( A < 0 \/ 0 < A ) ) )
10 4 9 bitr4d 
 |-  ( ( A e. RR /\ 0 <_ A ) -> ( A =/= 0 <-> 0 < A ) )