Theorem rexn0

Description: Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007) Avoid df-clel , ax-8 . (Revised by Gino Giotto, 2-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		dfrex2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 )
2		rzal	⊢ ( 𝐴 = ∅ → ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 )
3	2	con3i	⊢ ( ¬ ∀ 𝑥 ∈ 𝐴 ¬ 𝜑 → ¬ 𝐴 = ∅ )
4	1 3	sylbi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ¬ 𝐴 = ∅ )
5	4	neqned	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 → 𝐴 ≠ ∅ )