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Metamath Proof Explorer

Theorem uzubico2

Description: The upper integers are unbounded above. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypotheses uzubico2.1 
|- ( ph -> M e. ZZ )
uzubico2.2 
|- Z = ( ZZ>= ` M )
Assertion uzubico2 
|- ( ph -> A. x e. RR E. k e. ( x [,) +oo ) k e. Z )

Proof

Step Hyp Ref Expression
1 uzubico2.1 
 |-  ( ph -> M e. ZZ )
2 uzubico2.2 
 |-  Z = ( ZZ>= ` M )
3 1 2 uzubioo2 
 |-  ( ph -> A. x e. RR E. k e. ( x (,) +oo ) k e. Z )
4 ioossico 
 |-  ( x (,) +oo ) C_ ( x [,) +oo )
5 ssrexv 
 |-  ( ( x (,) +oo ) C_ ( x [,) +oo ) -> ( E. k e. ( x (,) +oo ) k e. Z -> E. k e. ( x [,) +oo ) k e. Z ) )
6 4 5 ax-mp 
 |-  ( E. k e. ( x (,) +oo ) k e. Z -> E. k e. ( x [,) +oo ) k e. Z )
7 6 ralimi 
 |-  ( A. x e. RR E. k e. ( x (,) +oo ) k e. Z -> A. x e. RR E. k e. ( x [,) +oo ) k e. Z )
8 3 7 syl 
 |-  ( ph -> A. x e. RR E. k e. ( x [,) +oo ) k e. Z )