Metamath Proof Explorer

Theorem nn0negleid

Description: A nonnegative integer is greater than or equal to its negative. (Contributed by AV, 13-Aug-2021)

Ref Expression
Assertion nn0negleid 
|- ( A e. NN0 -> -u A <_ A )

Proof

Step Hyp Ref Expression
1 nn0re 
 |-  ( A e. NN0 -> A e. RR )
2 1 renegcld 
 |-  ( A e. NN0 -> -u A e. RR )
3 0red 
 |-  ( A e. NN0 -> 0 e. RR )
4 nn0ge0 
 |-  ( A e. NN0 -> 0 <_ A )
5 1 le0neg2d 
 |-  ( A e. NN0 -> ( 0 <_ A <-> -u A <_ 0 ) )
6 4 5 mpbid 
 |-  ( A e. NN0 -> -u A <_ 0 )
7 2 3 1 6 4 letrd 
 |-  ( A e. NN0 -> -u A <_ A )