Addition of Cardinals Preserves Ordering - kipawaa/Proof-Tree GitHub Wiki

Statement

If $\kappa_1 = \lvert A_1 \rvert, \lambda_1 = \lvert B_1 \rvert, \kappa_2 = \lvert A_2 \rvert, \lambda_2 = \lvert B_2 \rvert$ such that $A_1 \cap B_1 = \emptyset$ and $A_2 \cap B_2 = \emptyset$ such that $\kappa_1 \leq \kappa_2$ and $\lambda_1 \leq \lambda_2$, then $\kappa_1 + \lambda_1 \leq \kappa_2 + \lambda_2$.

Explanation

Proof(s)

Since $\kappa_1 \leq \kappa_2$ we have that there exists an Injective Function $f : A_1 \to A_2$ by the definiton of Ordering of Cardinal Numbers.
Since $\lambda_1 \leq \lambda_2$ we have that there exists an Injective Function $g : B_1 \to B_2$ by the definition of Ordering of Cardinal Numbers.

We define the function $h : A_1 \cup B_1 \to A_2 \cup B_2$ such that

$$h(x) = \begin{cases} f(x) & x \in A_1\ g(x) & x \in B_1 \end{cases}$$

Since $A_1$ and $B_1$ are Disjoint Sets we have that $h$ is a Function.
Since $A_2, B_2$ are [Disjoint Sets]] we have that $h$ is [Injective since $f$ and $g$ are Injective.
Hence we have that $\lvert A_1 \cup B_1 \rvert \leq \lvert A_2 \cup B_2 \rvert$ by the definition of Ordering of Cardinal Numbers.
Hence $\kappa_1 + \lambda_1 \leq \kappa_2 + \lambda_2$, as wanted.

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