Relation between implication and subset relation If $a \implies b$ can we then generally say that $a \subseteq b$ ? 
For example: 
if $a: x > 15$ and $b: x >10$ then clearly $a \implies b$ and if we look at the sets represented by a $\lbrace16,17,18..\rbrace$ and b $\lbrace 11,12,13... \rbrace$ it is also obvious that $a \subset b$.

No but you can say:
Let P be (x ∈ A)
Let Q be (x ∈ B)
A ⊆ B ≡ ∀x, P → Q 

⊆ is for sets
→ is for propositions 

Note: You cannot say the same for A ⊂ B
Why?
Let A be {1} 
Let B be {1} 
Let P be (x ∈ A)
Let Q be (x ∈ B)
A ⊂ B is false.
∀x, P → Q is true.
Therefore A ⊂ B cannot be equivalent to ∀x, P → Q
