Theorem madeun

Description: The made set is the union of the old set and the new set. (Contributed by Scott Fenton, 8-Aug-2024)

	Ref	Expression
Assertion	madeun	$$⊢A\in \mathrm{On}\to M\left(A\right)=\mathrm{Old}\left(A\right)\cup N\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		newval	$$⊢A\in \mathrm{On}\to N\left(A\right)=M\left(A\right)\setminus \mathrm{Old}\left(A\right)$$
2	1	uneq2d	$$⊢A\in \mathrm{On}\to \mathrm{Old}\left(A\right)\cup N\left(A\right)=\mathrm{Old}\left(A\right)\cup \left(M\left(A\right)\setminus \mathrm{Old}\left(A\right)\right)$$
3		oldssmade	$$⊢A\in \mathrm{On}\to \mathrm{Old}\left(A\right)\subseteq M\left(A\right)$$
4		undif	$$⊢\mathrm{Old}\left(A\right)\subseteq M\left(A\right)\leftrightarrow \mathrm{Old}\left(A\right)\cup \left(M\left(A\right)\setminus \mathrm{Old}\left(A\right)\right)=M\left(A\right)$$
5	3 4	sylib	$$⊢A\in \mathrm{On}\to \mathrm{Old}\left(A\right)\cup \left(M\left(A\right)\setminus \mathrm{Old}\left(A\right)\right)=M\left(A\right)$$
6	2 5	eqtr2d	$$⊢A\in \mathrm{On}\to M\left(A\right)=\mathrm{Old}\left(A\right)\cup N\left(A\right)$$