Metamath Proof Explorer

Theorem nnsubi

Description: Subtraction of positive integers. (Contributed by NM, 19-Aug-2001)

Ref Expression
Hypotheses nnsub.1 
|- A e. NN
nnsub.2 
|- B e. NN
Assertion nnsubi 
|- ( A < B <-> ( B - A ) e. NN )

Proof

Step Hyp Ref Expression
1 nnsub.1 
 |-  A e. NN
2 nnsub.2 
 |-  B e. NN
3 nnsub 
 |-  ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( B - A ) e. NN ) )
4 1 2 3 mp2an 
 |-  ( A < B <-> ( B - A ) e. NN )