Theorem alscn0d

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)

	Ref	Expression
Hypothesis	alscn0d.1	⊢ ( 𝜑 → ∀! 𝑥 ∈ 𝐴 𝜓 )
Assertion	alscn0d	⊢ ( 𝜑 → 𝐴 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		alscn0d.1	⊢ ( 𝜑 → ∀! 𝑥 ∈ 𝐴 𝜓 )
2	1	alsc2d	⊢ ( 𝜑 → ∃ 𝑥 𝑥 ∈ 𝐴 )
3		n0	⊢ ( 𝐴 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ 𝐴 )
4	2 3	sylibr	⊢ ( 𝜑 → 𝐴 ≠ ∅ )