Theorem rmotru

Description: Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024) (Proof shortened by BJ, 23-Sep-2024)

	Ref	Expression
Assertion	rmotru	$$⊢{\exists }^{*}xx\in A\leftrightarrow {\exists }^{*}x\in A\top$$

Proof

Step	Hyp	Ref	Expression
1		tru	$$⊢\top$$
2	1	biantru	$$⊢x\in A\leftrightarrow \left(x\in A\wedge \top \right)$$
3	2	mobii	$$⊢{\exists }^{*}xx\in A\leftrightarrow {\exists }^{*}x\left(x\in A\wedge \top \right)$$
4		df-rmo	$$⊢{\exists }^{*}x\in A\top \leftrightarrow {\exists }^{*}x\left(x\in A\wedge \top \right)$$
5	3 4	bitr4i	$$⊢{\exists }^{*}xx\in A\leftrightarrow {\exists }^{*}x\in A\top$$