reusn

Description: A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006) (Proof shortened by Mario Carneiro, 14-Nov-2016)

		Ref	Expression
	Assertion	reusn	$⊢ \exists! x \in A φ \leftrightarrow \exists y \{x \in A \| φ\} = \{y\}$

Proof

Step	Hyp	Ref	Expression
1		euabsn2	$⊢ \exists! x (x \in A \land φ) \leftrightarrow \exists y \{x \| (x \in A \land φ)\} = \{y\}$
2		df-reu	$⊢ \exists! x \in A φ \leftrightarrow \exists! x (x \in A \land φ)$
3		df-rab	$⊢ \{x \in A \| φ\} = \{x \| (x \in A \land φ)\}$
4	3	eqeq1i	$⊢ \{x \in A \| φ\} = \{y\} \leftrightarrow \{x \| (x \in A \land φ)\} = \{y\}$
5	4	exbii	$⊢ \exists y \{x \in A \| φ\} = \{y\} \leftrightarrow \exists y \{x \| (x \in A \land φ)\} = \{y\}$
6	1 2 5	3bitr4i	$⊢ \exists! x \in A φ \leftrightarrow \exists y \{x \in A \| φ\} = \{y\}$

Metamath Proof Explorer

Theorem reusn

Proof