Description: The infimum of a nonempty bounded set of reals is the greatest lower bound. (Contributed by Glauco Siliprandi, 29-Jun-2017) (Revised by AV, 15-Sep-2020)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | infrglb | |- ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A x <_ y ) /\ B e. RR ) -> ( inf ( A , RR , < ) < B <-> E. z e. A z < B ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltso | |- < Or RR |
|
| 2 | 1 | a1i | |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A x <_ y ) -> < Or RR ) |
| 3 | infm3 | |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A x <_ y ) -> E. x e. RR ( A. y e. A -. y < x /\ A. y e. RR ( x < y -> E. z e. A z < y ) ) ) |
|
| 4 | simp1 | |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A x <_ y ) -> A C_ RR ) |
|
| 5 | 2 3 4 | infglbb | |- ( ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A x <_ y ) /\ B e. RR ) -> ( inf ( A , RR , < ) < B <-> E. z e. A z < B ) ) |