Theorem axprlem2 5318

Description: Lemma for axpr 5322. There exists a set to which all sets whose only members are empty sets belong. (Contributed by Rohan Ridenour, 9-Aug-2023.) (Revised by BJ, 13-Aug-2023.)

Assertion

Ref	Expression
axprlem2	⊢ ∃𝑥∀𝑦(∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥)

Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem axprlem2

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pow 5259	. . 3 ⊢ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑣) → 𝑦 ∈ 𝑥)
2		df-ral 3142	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 ↔ ∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤 ¬ 𝑤 ∈ 𝑧))
3		imim2 58	. . . . . . . 8 ⊢ ((∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑧 ∈ 𝑣) → ((𝑧 ∈ 𝑦 → ∀𝑤 ¬ 𝑤 ∈ 𝑧) → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑣)))
4	3	al2imi 1815	. . . . . . 7 ⊢ (∀𝑧(∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑧 ∈ 𝑣) → (∀𝑧(𝑧 ∈ 𝑦 → ∀𝑤 ¬ 𝑤 ∈ 𝑧) → ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑣)))
5	2, 4	syl5bi 244	. . . . . 6 ⊢ (∀𝑧(∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑧 ∈ 𝑣) → (∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → ∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑣)))
6	5	imim1d 82	. . . . 5 ⊢ (∀𝑧(∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑧 ∈ 𝑣) → ((∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑣) → 𝑦 ∈ 𝑥) → (∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥)))
7	6	alimdv 1916	. . . 4 ⊢ (∀𝑧(∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑧 ∈ 𝑣) → (∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑣) → 𝑦 ∈ 𝑥) → ∀𝑦(∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥)))
8	7	eximdv 1917	. . 3 ⊢ (∀𝑧(∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑧 ∈ 𝑣) → (∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑣) → 𝑦 ∈ 𝑥) → ∃𝑥∀𝑦(∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥)))
9	1, 8	mpi 20	. 2 ⊢ (∀𝑧(∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑧 ∈ 𝑣) → ∃𝑥∀𝑦(∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥))
10		axprlem1 5317	. 2 ⊢ ∃𝑣∀𝑧(∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑧 ∈ 𝑣)
11	9, 10	exlimiiv 1931	1 ⊢ ∃𝑥∀𝑦(∀𝑧 ∈ 𝑦 ∀𝑤 ¬ 𝑤 ∈ 𝑧 → 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1534  ∃wex 1779  ∀wral 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-nul 5203  ax-pow 5259

This theorem depends on definitions:  df-bi 209  df-ex 1780  df-ral 3142

This theorem is referenced by:  axpr  5322