Description: A nonempty finite set of real numbers is bounded (image set version). (Contributed by Glauco Siliprandi, 23-Oct-2021)
Ref | Expression | ||
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Hypotheses | fimaxre4.1 | |- F/ x ph |
|
fimaxre4.2 | |- ( ph -> A e. Fin ) |
||
fimaxre4.3 | |- ( ( ph /\ x e. A ) -> B e. RR ) |
||
Assertion | fimaxre4 | |- ( ph -> E. y e. RR A. x e. A B <_ y ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fimaxre4.1 | |- F/ x ph |
|
2 | fimaxre4.2 | |- ( ph -> A e. Fin ) |
|
3 | fimaxre4.3 | |- ( ( ph /\ x e. A ) -> B e. RR ) |
|
4 | 3 | ex | |- ( ph -> ( x e. A -> B e. RR ) ) |
5 | 1 4 | ralrimi | |- ( ph -> A. x e. A B e. RR ) |
6 | fimaxre3 | |- ( ( A e. Fin /\ A. x e. A B e. RR ) -> E. y e. RR A. x e. A B <_ y ) |
|
7 | 2 5 6 | syl2anc | |- ( ph -> E. y e. RR A. x e. A B <_ y ) |