Splitting up an or statement while proving subset

[smoky condor]

If I am trying to prove that set Xis a subset of set Y,

an i reach the point where for any x in X,
X is either in set A or set B.

can i simply choose x in set A, and derive that that implies x is in set Y? or do i need to do casework?

[smoky condor]

wait i asked that ina really confusing way

$$test$$

[mint tree]

edwardF8

[smoky condor]

okay wait lemme write this out better

[wanton marsh]

Hello ! If you can prove 𝐴⊆𝑌 and 𝐵⊆𝑌, then you’re done without explicit element-by-element casework !

[smoky condor]

$$\begin{align}
\text{Let x} \in A \
\text{through derivation}, x \in V \vee x \in W\
x \in V \implies x \in B (\text{through derivation}) \
\therefore A \subseteq B
\end{align}$$

[mint tree]

edwardF8
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[smoky condor]

ig this doesnt just have to be for subsets, but like if its a disjunction, can i jus take one of them and disregard the other or statement

[smoky condor]

okay so i have to address both cases

thank you!!!!