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Metamath Proof Explorer

Theorem eueq

Description: A class is a set if and only if there exists a unique set equal to it. (Contributed by NM, 25-Nov-1994) Shorten combined proofs of moeq and eueq . (Proof shortened by BJ, 24-Sep-2022)

		Ref	Expression
	Assertion	eueq	⊢ ( 𝐴 ∈ V ↔ ∃! 𝑥 𝑥 = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		moeq	⊢ ∃* 𝑥 𝑥 = 𝐴
2	1	biantru	⊢ ( ∃ 𝑥 𝑥 = 𝐴 ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃* 𝑥 𝑥 = 𝐴 ) )
3		isset	⊢ ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 )
4		df-eu	⊢ ( ∃! 𝑥 𝑥 = 𝐴 ↔ ( ∃ 𝑥 𝑥 = 𝐴 ∧ ∃* 𝑥 𝑥 = 𝐴 ) )
5	2 3 4	3bitr4i	⊢ ( 𝐴 ∈ V ↔ ∃! 𝑥 𝑥 = 𝐴 )