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Metamath Proof Explorer

Theorem uzssico

Description: Upper integer sets are a subset of the corresponding closed-below, open-above intervals. (Contributed by Thierry Arnoux, 29-Dec-2021)

Ref Expression
Assertion uzssico 
|- ( M e. ZZ -> ( ZZ>= ` M ) C_ ( M [,) +oo ) )

Proof

Step Hyp Ref Expression
1 zssre 
 |-  ZZ C_ RR
2 1 sseli 
 |-  ( x e. ZZ -> x e. RR )
3 2 a1i 
 |-  ( M e. ZZ -> ( x e. ZZ -> x e. RR ) )
4 3 anim1d 
 |-  ( M e. ZZ -> ( ( x e. ZZ /\ M <_ x ) -> ( x e. RR /\ M <_ x ) ) )
5 eluz1 
 |-  ( M e. ZZ -> ( x e. ( ZZ>= ` M ) <-> ( x e. ZZ /\ M <_ x ) ) )
6 zre 
 |-  ( M e. ZZ -> M e. RR )
7 elicopnf 
 |-  ( M e. RR -> ( x e. ( M [,) +oo ) <-> ( x e. RR /\ M <_ x ) ) )
8 6 7 syl 
 |-  ( M e. ZZ -> ( x e. ( M [,) +oo ) <-> ( x e. RR /\ M <_ x ) ) )
9 4 5 8 3imtr4d 
 |-  ( M e. ZZ -> ( x e. ( ZZ>= ` M ) -> x e. ( M [,) +oo ) ) )
10 9 ssrdv 
 |-  ( M e. ZZ -> ( ZZ>= ` M ) C_ ( M [,) +oo ) )