Theorem hashelne0d

Description: A set with an element has nonzero size. (Contributed by Rohan Ridenour, 3-Aug-2023)

	Ref	Expression
Hypotheses	hashelne0d.1	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
	hashelne0d.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
Assertion	hashelne0d	⊢ ( 𝜑 → ¬ ( ♯ ‘ 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		hashelne0d.1	⊢ ( 𝜑 → 𝐵 ∈ 𝐴 )
2		hashelne0d.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
3	1	ne0d	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
4	3	neneqd	⊢ ( 𝜑 → ¬ 𝐴 = ∅ )
5		hasheq0	⊢ ( 𝐴 ∈ 𝑉 → ( ( ♯ ‘ 𝐴 ) = 0 ↔ 𝐴 = ∅ ) )
6	2 5	syl	⊢ ( 𝜑 → ( ( ♯ ‘ 𝐴 ) = 0 ↔ 𝐴 = ∅ ) )
7	4 6	mtbird	⊢ ( 𝜑 → ¬ ( ♯ ‘ 𝐴 ) = 0 )