Description: "Less than" implies "less than or equal to". ( pssss analog.) (Contributed by NM, 4-Dec-2011)
Ref | Expression | ||
---|---|---|---|
Hypotheses | pltval.l | |- .<_ = ( le ` K ) |
|
pltval.s | |- .< = ( lt ` K ) |
||
Assertion | pltle | |- ( ( K e. A /\ X e. B /\ Y e. C ) -> ( X .< Y -> X .<_ Y ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltval.l | |- .<_ = ( le ` K ) |
|
2 | pltval.s | |- .< = ( lt ` K ) |
|
3 | 1 2 | pltval | |- ( ( K e. A /\ X e. B /\ Y e. C ) -> ( X .< Y <-> ( X .<_ Y /\ X =/= Y ) ) ) |
4 | 3 | simprbda | |- ( ( ( K e. A /\ X e. B /\ Y e. C ) /\ X .< Y ) -> X .<_ Y ) |
5 | 4 | ex | |- ( ( K e. A /\ X e. B /\ Y e. C ) -> ( X .< Y -> X .<_ Y ) ) |