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Metamath Proof Explorer

Theorem fznn

Description: Finite set of sequential integers starting at 1. (Contributed by NM, 31-Aug-2011) (Revised by Mario Carneiro, 18-Jun-2015)

Ref Expression
Assertion fznn 
|- ( N e. ZZ -> ( K e. ( 1 ... N ) <-> ( K e. NN /\ K <_ N ) ) )

Proof

Step Hyp Ref Expression
1 elfzuzb 
 |-  ( K e. ( 1 ... N ) <-> ( K e. ( ZZ>= ` 1 ) /\ N e. ( ZZ>= ` K ) ) )
2 elnnuz 
 |-  ( K e. NN <-> K e. ( ZZ>= ` 1 ) )
3 2 anbi1i 
 |-  ( ( K e. NN /\ N e. ( ZZ>= ` K ) ) <-> ( K e. ( ZZ>= ` 1 ) /\ N e. ( ZZ>= ` K ) ) )
4 1 3 bitr4i 
 |-  ( K e. ( 1 ... N ) <-> ( K e. NN /\ N e. ( ZZ>= ` K ) ) )
5 nnz 
 |-  ( K e. NN -> K e. ZZ )
6 eluz 
 |-  ( ( K e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` K ) <-> K <_ N ) )
7 5 6 sylan 
 |-  ( ( K e. NN /\ N e. ZZ ) -> ( N e. ( ZZ>= ` K ) <-> K <_ N ) )
8 7 ancoms 
 |-  ( ( N e. ZZ /\ K e. NN ) -> ( N e. ( ZZ>= ` K ) <-> K <_ N ) )
9 8 pm5.32da 
 |-  ( N e. ZZ -> ( ( K e. NN /\ N e. ( ZZ>= ` K ) ) <-> ( K e. NN /\ K <_ N ) ) )
10 4 9 bitrid 
 |-  ( N e. ZZ -> ( K e. ( 1 ... N ) <-> ( K e. NN /\ K <_ N ) ) )