# Big Union Big Intersection Proof

• May 5th 2011, 03:48 AM
StaryNight
Big Union Big Intersection Proof
I'm having trouble working out how to prove the following proposition:

If A and B are non-empty and ∀X∈A.∀Y∈B. X⊆Y then ∐A ⊆ ∩B.

I know that ∐A = {x|∃X ∈A. x∈X} and ∩B = {y|∀Y∈B. y∈Y} and to show the subset relation holds I need to show that x∈∐A ⇒ x∈∩B. I feel like I understand why the proposition has to hold intuitively but can't seem to come up with a proof.

$t \in \bigcup A \, \Rightarrow \,\left( {\exists T \in A} \right)\left[ {t \in T} \right]$
$\left( {\forall S \in B} \right)\left[ {T \subseteq S} \right]\; \Rightarrow \;\left( {\forall S \in B} \right)\left[ {t \in S} \right]$
$t\in \bigcap B$.