Theorem axprlem1OLD 5390

Description: Obsolete version of axprlem1 5385 as of 6-Apr-2026. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axprlem1OLD	⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axprlem1OLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pow 5327	. . 3 ⊢ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥)
2		pm2.21 124	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
3	2	alimi 1844	. . . . . . 7 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
4	3	a1i 11	. . . . . 6 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → (∀𝑧 ¬ 𝑧 ∈ 𝑦 → ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤)))
5	4	imim1d 83	. . . . 5 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → ((∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → (∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
6	5	alimdv 1949	. . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → (∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → ∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
7	6	eximdv 1950	. . 3 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → (∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
8	1, 7	mpi 21	. 2 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥))
9		ax-nul 5260	. 2 ⊢ ∃𝑤∀𝑧 ¬ 𝑧 ∈ 𝑤
10	8, 9	exlimiiv 1964	1 ⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-nul 5260  ax-pow 5327

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by: (None)