Theorem axprlem1 5346

Description: Lemma for axpr 5351. There exists a set to which all empty sets belong. (Contributed by Rohan Ridenour, 10-Aug-2023.) (Revised by BJ, 13-Aug-2023.)

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 5288	. . 3 ⊢ ∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥)
2		pm2.21 123	. . . . . . . 8 ⊢ (¬ 𝑧 ∈ 𝑦 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤))
3	2	alimi 1814	. . . . . . 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 82	. . . . 5 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → ((∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → (∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
6	5	alimdv 1919	. . . 4 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → (∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → ∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
7	6	eximdv 1920	. . 3 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → (∃𝑥∀𝑦(∀𝑧(𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑤) → 𝑦 ∈ 𝑥) → ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)))
8	1, 7	mpi 20	. 2 ⊢ (∀𝑧 ¬ 𝑧 ∈ 𝑤 → ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥))
9		ax-nul 5230	. 2 ⊢ ∃𝑤∀𝑧 ¬ 𝑧 ∈ 𝑤
10	8, 9	exlimiiv 1934	1 ⊢ ∃𝑥∀𝑦(∀𝑧 ¬ 𝑧 ∈ 𝑦 → 𝑦 ∈ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-nul 5230  ax-pow 5288

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by:  axprlem2  5347  axprlem4  5349