Theorem axprlem1

Description: Lemma for axpr . There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023) (Revised by BJ, 13-Aug-2023) (Proof shortened by Matthew House, 6-Apr-2026)

	Ref	Expression
Assertion	axprlem1	⊢ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ax-pow	⊢ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤 ) → 𝑦 ∈ 𝑥 )
2		pm2.21	⊢ ( ¬ 𝑧 ∈ 𝑦 → ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤 ) )
3	2	alimi	⊢ ( ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 → ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤 ) )
4	3	imim1i	⊢ ( ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤 ) → 𝑦 ∈ 𝑥 ) → ( ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥 ) )
5	4	alimi	⊢ ( ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤 ) → 𝑦 ∈ 𝑥 ) → ∀ 𝑦 ( ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥 ) )
6	1 5	eximii	⊢ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥 )