Theorem euhash1

Description: The size of a set is 1 in terms of existential uniqueness. (Contributed by Alexander van der Vekens, 8-Feb-2018)

	Ref	Expression
Assertion	euhash1	⊢ ( 𝑉 ∈ 𝑊 → ( ( ♯ ‘ 𝑉 ) = 1 ↔ ∃! 𝑎 𝑎 ∈ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		hashen1	⊢ ( 𝑉 ∈ 𝑊 → ( ( ♯ ‘ 𝑉 ) = 1 ↔ 𝑉 ≈ 1o ) )
2		euen1b	⊢ ( 𝑉 ≈ 1o ↔ ∃! 𝑎 𝑎 ∈ 𝑉 )
3	1 2	bitrdi	⊢ ( 𝑉 ∈ 𝑊 → ( ( ♯ ‘ 𝑉 ) = 1 ↔ ∃! 𝑎 𝑎 ∈ 𝑉 ) )