Theorem onunisuc

Description: An ordinal number is equal to the union of its successor. (Contributed by NM, 12-Jun-1994) Generalize from onunisuci . (Revised by BJ, 28-Dec-2024)

	Ref	Expression
Assertion	onunisuc	$$⊢A\in \mathrm{On}\to \bigcup \mathrm{suc}A=A$$

Proof

Step	Hyp	Ref	Expression
1		ontr	$$⊢A\in \mathrm{On}\to \mathrm{Tr}A$$
2		unisucg	$$⊢A\in \mathrm{On}\to \left(\mathrm{Tr}A\leftrightarrow \bigcup \mathrm{suc}A=A\right)$$
3	1 2	mpbid	$$⊢A\in \mathrm{On}\to \bigcup \mathrm{suc}A=A$$