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Metamath Proof Explorer

Theorem uzubioo2

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

Ref Expression
Hypotheses uzubioo2.1 
|- ( ph -> M e. ZZ )
uzubioo2.2 
|- Z = ( ZZ>= ` M )
Assertion uzubioo2 
|- ( ph -> A. x e. RR E. k e. ( x (,) +oo ) k e. Z )

Proof

Step Hyp Ref Expression
1 uzubioo2.1 
 |-  ( ph -> M e. ZZ )
2 uzubioo2.2 
 |-  Z = ( ZZ>= ` M )
3 1 adantr 
 |-  ( ( ph /\ y e. RR ) -> M e. ZZ )
4 simpr 
 |-  ( ( ph /\ y e. RR ) -> y e. RR )
5 3 2 4 uzubioo 
 |-  ( ( ph /\ y e. RR ) -> E. k e. ( y (,) +oo ) k e. Z )
6 5 ralrimiva 
 |-  ( ph -> A. y e. RR E. k e. ( y (,) +oo ) k e. Z )
7 oveq1 
 |-  ( x = y -> ( x (,) +oo ) = ( y (,) +oo ) )
8 7 rexeqdv 
 |-  ( x = y -> ( E. k e. ( x (,) +oo ) k e. Z <-> E. k e. ( y (,) +oo ) k e. Z ) )
9 8 cbvralvw 
 |-  ( A. x e. RR E. k e. ( x (,) +oo ) k e. Z <-> A. y e. RR E. k e. ( y (,) +oo ) k e. Z )
10 6 9 sylibr 
 |-  ( ph -> A. x e. RR E. k e. ( x (,) +oo ) k e. Z )