Theorem ralsn0d

Description: Deduction rule: Given "all some" applied to a class, the class is not the empty set. (Contributed by David A. Wheeler, 23-Oct-2018) (Revised by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Hypothesis	ralsn0d.1	⊢ ( 𝜑 → ∀∃ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
Assertion	ralsn0d	⊢ ( 𝜑 → 𝐴 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		ralsn0d.1	⊢ ( 𝜑 → ∀∃ 𝑥 ∈ 𝐴 ( 𝜓 → 𝜒 ) )
2	1	rals2d	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
3		rexn0	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝐴 ≠ ∅ )
4	2 3	syl	⊢ ( 𝜑 → 𝐴 ≠ ∅ )