Set Difference is Set
Jump to navigation
Jump to search
Theorem
Let $x$ be a small class.
Let $A$ be a class.
Let $\map \MM B$ denote that $B$ is small.
Then:
- $\map \MM {x \setminus A}$
Proof
This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.
If you see any proofs that link to this page, please insert this template at the top.
If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.
By Set Difference as Intersection with Relative Complement:
- $\paren {x \setminus A} = \paren {x \cap \map \complement A}$
Work In Progress In particular: This rule must be proven for classes You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code. |
Next, by Axiom of Subsets Equivalents, $\paren {x \cap \map \complement A}$ is small.
Therefore:
- $\map \MM {x \setminus A}$
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 5.15$