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 )