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Metamath Proof Explorer

Theorem euen1

Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014)

		Ref	Expression
	Assertion	euen1	$⊢ \exists! x φ \leftrightarrow \{x \| φ\} \approx 1_{𝑜}$

Step	Hyp	Ref	Expression
1		reuen1	$⊢ \exists! x \in V φ \leftrightarrow \{x \in V \| φ\} \approx 1_{𝑜}$
2		reuv	$⊢ \exists! x \in V φ \leftrightarrow \exists! x φ$
3		rabab	$⊢ \{x \in V \| φ\} = \{x \| φ\}$
4	3	breq1i	$⊢ \{x \in V \| φ\} \approx 1_{𝑜} \leftrightarrow \{x \| φ\} \approx 1_{𝑜}$
5	1 2 4	3bitr3i	$⊢ \exists! x φ \leftrightarrow \{x \| φ\} \approx 1_{𝑜}$