Theorem rexn0

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007)

	Ref	Expression
Assertion	rexn0	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		ne0i	⊢ ( 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅ )
2	1	a1d	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝐴 ≠ ∅ ) )
3	2	rexlimiv	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅ )