Theorem euen1b

Description: Two ways to express " A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015)

	Ref	Expression
Assertion	euen1b	$$⊢A\approx 1_{𝑜}\leftrightarrow \exists !xx\in A$$

Proof

Step	Hyp	Ref	Expression
1		euen1	$$⊢\exists !xx\in A\leftrightarrow \left\{x|x\in A\right\}\approx 1_{𝑜}$$
2		abid2	$$⊢\left\{x|x\in A\right\}=A$$
3	2	breq1i	$$⊢\left\{x|x\in A\right\}\approx 1_{𝑜}\leftrightarrow A\approx 1_{𝑜}$$
4	1 3	bitr2i	$$⊢A\approx 1_{𝑜}\leftrightarrow \exists !xx\in A$$