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Metamath Proof Explorer

Theorem fzossfzop1

Description: A half-open range of nonnegative integers is a subset of a half-open range of nonnegative integers with the upper bound increased by one. (Contributed by Alexander van der Vekens, 5-Aug-2018)

Ref Expression
Assertion fzossfzop1 
|- ( N e. NN0 -> ( 0 ..^ N ) C_ ( 0 ..^ ( N + 1 ) ) )

Proof

Step Hyp Ref Expression
1 nn0z 
 |-  ( N e. NN0 -> N e. ZZ )
2 id 
 |-  ( N e. ZZ -> N e. ZZ )
3 peano2z 
 |-  ( N e. ZZ -> ( N + 1 ) e. ZZ )
4 zre 
 |-  ( N e. ZZ -> N e. RR )
5 4 lep1d 
 |-  ( N e. ZZ -> N <_ ( N + 1 ) )
6 2 3 5 3jca 
 |-  ( N e. ZZ -> ( N e. ZZ /\ ( N + 1 ) e. ZZ /\ N <_ ( N + 1 ) ) )
7 1 6 syl 
 |-  ( N e. NN0 -> ( N e. ZZ /\ ( N + 1 ) e. ZZ /\ N <_ ( N + 1 ) ) )
8 eluz2 
 |-  ( ( N + 1 ) e. ( ZZ>= ` N ) <-> ( N e. ZZ /\ ( N + 1 ) e. ZZ /\ N <_ ( N + 1 ) ) )
9 7 8 sylibr 
 |-  ( N e. NN0 -> ( N + 1 ) e. ( ZZ>= ` N ) )
10 fzoss2 
 |-  ( ( N + 1 ) e. ( ZZ>= ` N ) -> ( 0 ..^ N ) C_ ( 0 ..^ ( N + 1 ) ) )
11 9 10 syl 
 |-  ( N e. NN0 -> ( 0 ..^ N ) C_ ( 0 ..^ ( N + 1 ) ) )