Theorem nrexrmo

Description: Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017)

	Ref	Expression
Assertion	nrexrmo	⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 𝜑 → ∃* 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		pm2.21	⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜑 ) )
2		rmo5	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃! 𝑥 ∈ 𝐴 𝜑 ) )
3	1 2	sylibr	⊢ ( ¬ ∃ 𝑥 ∈ 𝐴 𝜑 → ∃* 𝑥 ∈ 𝐴 𝜑 )