Metamath Proof Explorer


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 ( 𝑉𝑊 → ( ( ♯ ‘ 𝑉 ) = 1 ↔ ∃! 𝑎 𝑎𝑉 ) )

Proof

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