Theorem reutru

Description: Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024)

	Ref	Expression
Assertion	reutru	$$⊢\exists !xx\in A\leftrightarrow \exists !x\in A\top$$

Proof

Step	Hyp	Ref	Expression
1		tru	$$⊢\top$$
2	1	biantru	$$⊢x\in A\leftrightarrow \left(x\in A\wedge \top \right)$$
3	2	eubii	$$⊢\exists !xx\in A\leftrightarrow \exists !x\left(x\in A\wedge \top \right)$$
4		df-reu	$$⊢\exists !x\in A\top \leftrightarrow \exists !x\left(x\in A\wedge \top \right)$$
5	3 4	bitr4i	$$⊢\exists !xx\in A\leftrightarrow \exists !x\in A\top$$