Theorem reutru

Description: Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024)

	Ref	Expression
Assertion	reutru	⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑥 ∈ 𝐴 ⊤ )

Proof

Step	Hyp	Ref	Expression
1		tru	⊢ ⊤
2	1	biantru	⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ ⊤ ) )
3	2	eubii	⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ⊤ ) )
4		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 ⊤ ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ ⊤ ) )
5	3 4	bitr4i	⊢ ( ∃! 𝑥 𝑥 ∈ 𝐴 ↔ ∃! 𝑥 ∈ 𝐴 ⊤ )