Theorem axnultcoreg 36668

Description: Derivation of ax-nul 5241 from ax-tco 36660 and ax-reg 9498. Use ax-nul 5241 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axnultcoreg	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem axnultcoreg

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ1 2121	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
2	1	biimprd 248	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑧 ∈ 𝑤 → 𝑥 ∈ 𝑤))
3	2	spimevw 1987	. . . 4 ⊢ (𝑧 ∈ 𝑤 → ∃𝑥 𝑥 ∈ 𝑤)
4		ax-reg 9498	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝑤 → ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤)))
5	3, 4	syl 17	. . 3 ⊢ (𝑧 ∈ 𝑤 → ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤)))
6		pm2.65 193	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤) → ((𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤) → ¬ 𝑦 ∈ 𝑥))
7	6	al2imi 1817	. . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤) → (∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤) → ∀𝑦 ¬ 𝑦 ∈ 𝑥))
8	7	imim2i 16	. . . . 5 ⊢ ((𝑥 ∈ 𝑤 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤)) → (𝑥 ∈ 𝑤 → (∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤) → ∀𝑦 ¬ 𝑦 ∈ 𝑥)))
9	8	impd 410	. . . 4 ⊢ ((𝑥 ∈ 𝑤 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤)) → ((𝑥 ∈ 𝑤 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤)) → ∀𝑦 ¬ 𝑦 ∈ 𝑥))
10	9	aleximi 1834	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝑤 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤)) → (∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤)) → ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥))
11	5, 10	mpan9 506	. 2 ⊢ ((𝑧 ∈ 𝑤 ∧ ∀𝑥(𝑥 ∈ 𝑤 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤))) → ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
12		ax-tco 36660	. 2 ⊢ ∃𝑤(𝑧 ∈ 𝑤 ∧ ∀𝑥(𝑥 ∈ 𝑤 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤)))
13	11, 12	exlimiiv 1933	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀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-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-reg 9498  ax-tco 36660

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by: (None)