Theorem axprlem1 5360

Description: Lemma for axpr 5364. 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.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axprlem1

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pow 5302	. 2 ⊢ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥)
2		pm2.21 123	. . . . 5 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
3	2	alimi 1813	. . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
4	3	imim1i 63	. . 3 ⊢ ((∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → (∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥))
5	4	alimi 1813	. 2 ⊢ (∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → ∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥))
6	1, 5	eximii 1839	1 ⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-pow 5302

This theorem depends on definitions:  df-bi 207  df-ex 1782

This theorem is referenced by:  axprlem2  5361  axpr  5364  axprlem4OLD  5367