Theorem dfom6

Description: Let _om be defined to be the union of the set of all finite ordinals. (Contributed by RP, 27-Sep-2023)

	Ref	Expression
Assertion	dfom6	$$⊢\mathrm{\omega }=\bigcup \left(\mathrm{On}\cap \mathrm{Fin}\right)$$

Proof

Step	Hyp	Ref	Expression
1		limom	$$⊢\mathrm{Lim}\mathrm{\omega }$$
2		limuni	$$⊢\mathrm{Lim}\mathrm{\omega }\to \mathrm{\omega }=\bigcup \mathrm{\omega }$$
3	1 2	ax-mp	$$⊢\mathrm{\omega }=\bigcup \mathrm{\omega }$$
4		onfin2	$$⊢\mathrm{\omega }=\mathrm{On}\cap \mathrm{Fin}$$
5	4	unieqi	$$⊢\bigcup \mathrm{\omega }=\bigcup \left(\mathrm{On}\cap \mathrm{Fin}\right)$$
6	3 5	eqtri	$$⊢\mathrm{\omega }=\bigcup \left(\mathrm{On}\cap \mathrm{Fin}\right)$$