Theorem elpsseu

Description: A class has a unique element just in case it has a unique proper subset. (Contributed by Adrian Ducourtial, 26-Oct-2026)

	Ref	Expression
Assertion	elpsseu	⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑦 𝑦 ⊊ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		elpssex	⊢ ( ∃ 𝑥 𝑥 ∈ 𝐴 ↔ ∃ 𝑦 𝑦 ⊊ 𝐴 )
2		elpssmo	⊢ ( ∃* 𝑥 𝑥 ∈ 𝐴 ↔ ∃* 𝑦 𝑦 ⊊ 𝐴 )
3	1 2	anbi12i	⊢ ( ( ∃ 𝑥 𝑥 ∈ 𝐴 ∧ ∃* 𝑥 𝑥 ∈ 𝐴 ) ↔ ( ∃ 𝑦 𝑦 ⊊ 𝐴 ∧ ∃* 𝑦 𝑦 ⊊ 𝐴 ) )
4		df-eu	⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ( ∃ 𝑥 𝑥 ∈ 𝐴 ∧ ∃* 𝑥 𝑥 ∈ 𝐴 ) )
5		df-eu	⊢ ( ∃! 𝑦 𝑦 ⊊ 𝐴 ↔ ( ∃ 𝑦 𝑦 ⊊ 𝐴 ∧ ∃* 𝑦 𝑦 ⊊ 𝐴 ) )
6	3 4 5	3bitr4i	⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑦 𝑦 ⊊ 𝐴 )