Metamath Proof Explorer

Theorem elznn

Description: Integer property expressed in terms of positive integers and nonnegative integers. (Contributed by NM, 12-Jul-2005)

Ref Expression
Assertion elznn 
|- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ -u N e. NN0 ) ) )

Proof

Step Hyp Ref Expression
1 elz 
 |-  ( N e. ZZ <-> ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) )
2 3orrot 
 |-  ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ -u N e. NN \/ N = 0 ) )
3 3orass 
 |-  ( ( N e. NN \/ -u N e. NN \/ N = 0 ) <-> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
4 2 3 bitri 
 |-  ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) )
5 elnn0 
 |-  ( -u N e. NN0 <-> ( -u N e. NN \/ -u N = 0 ) )
6 recn 
 |-  ( N e. RR -> N e. CC )
7 6 negeq0d 
 |-  ( N e. RR -> ( N = 0 <-> -u N = 0 ) )
8 7 orbi2d 
 |-  ( N e. RR -> ( ( -u N e. NN \/ N = 0 ) <-> ( -u N e. NN \/ -u N = 0 ) ) )
9 5 8 bitr4id 
 |-  ( N e. RR -> ( -u N e. NN0 <-> ( -u N e. NN \/ N = 0 ) ) )
10 9 orbi2d 
 |-  ( N e. RR -> ( ( N e. NN \/ -u N e. NN0 ) <-> ( N e. NN \/ ( -u N e. NN \/ N = 0 ) ) ) )
11 4 10 bitr4id 
 |-  ( N e. RR -> ( ( N = 0 \/ N e. NN \/ -u N e. NN ) <-> ( N e. NN \/ -u N e. NN0 ) ) )
12 11 pm5.32i 
 |-  ( ( N e. RR /\ ( N = 0 \/ N e. NN \/ -u N e. NN ) ) <-> ( N e. RR /\ ( N e. NN \/ -u N e. NN0 ) ) )
13 1 12 bitri 
 |-  ( N e. ZZ <-> ( N e. RR /\ ( N e. NN \/ -u N e. NN0 ) ) )