Theorem n0moeu

Description: A case of equivalence of "at most one" and "only one". (Contributed by FL, 6-Dec-2010)

	Ref	Expression
Assertion	n0moeu	⊢ ( 𝐴 ≠ ∅ → ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑥 𝑥 ∈ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		n0	⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )
2	1	biimpi	⊢ ( 𝐴 ≠ ∅ → ∃ 𝑥 𝑥 ∈ 𝐴 )
3	2	biantrurd	⊢ ( 𝐴 ≠ ∅ → ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ( ∃ 𝑥 𝑥 ∈ 𝐴 ∧ ∃* 𝑥 𝑥 ∈ 𝐴 ) ) )
4		df-eu	⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ( ∃ 𝑥 𝑥 ∈ 𝐴 ∧ ∃* 𝑥 𝑥 ∈ 𝐴 ) )
5	3 4	bitr4di	⊢ ( 𝐴 ≠ ∅ → ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑥 𝑥 ∈ 𝐴 ) )