Theorem axnulregtco 37086

Description: Derivation of ax-nul 5267 from ax-reg 9567 and ax-tco 37078. Use ax-nul 5267 instead. (Contributed by Matthew House, 7-Apr-2026.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Distinct variable group:   𝑥,𝑦

Proof of Theorem axnulregtco

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

Step	Hyp	Ref	Expression
1		elequ1 2152	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
2	1	biimprd 251	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑧 ∈ 𝑤 → 𝑥 ∈ 𝑤))
3	2	spimevw 2018	. . . 4 ⊢ (𝑧 ∈ 𝑤 → ∃𝑥 𝑥 ∈ 𝑤)
4		ax-reg 9567	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝑤 → ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤)))
5	3, 4	syl 18	. . 3 ⊢ (𝑧 ∈ 𝑤 → ∃𝑥(𝑥 ∈ 𝑤 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤)))
6		pm2.65 195	. . . . . . 7 ⊢ ((𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤) → ((𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤) → ¬ 𝑦 ∈ 𝑥))
7	6	al2imi 1848	. . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤) → (∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤) → ∀𝑦 ¬ 𝑦 ∈ 𝑥))
8	7	imim2i 17	. . . . 5 ⊢ ((𝑥 ∈ 𝑤 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤)) → (𝑥 ∈ 𝑤 → (∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤) → ∀𝑦 ¬ 𝑦 ∈ 𝑥)))
9	8	impd 416	. . . 4 ⊢ ((𝑥 ∈ 𝑤 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤)) → ((𝑥 ∈ 𝑤 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ¬ 𝑦 ∈ 𝑤)) → ∀𝑦 ¬ 𝑦 ∈ 𝑥))
10	9	aleximi 1865	. . 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 516	. 2 ⊢ ((𝑧 ∈ 𝑤 ∧ ∀𝑥(𝑥 ∈ 𝑤 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤))) → ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥)
12		ax-tco 37078	. 2 ⊢ ∃𝑤(𝑧 ∈ 𝑤 ∧ ∀𝑥(𝑥 ∈ 𝑤 → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑤)))
13	11, 12	exlimiiv 1964	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401  ∀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-6 2000  ax-7 2041  ax-8 2147  ax-reg 9567  ax-tco 37078

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)