Theorem axnulg 35870

Description: A generalization of ax-nul 5259 in which 𝑥 and 𝑦 need not be distinct. This theorem scheme bundles ax-nul 5259 with the degenerate instance ∃𝑥∀𝑥¬ 𝑥 ∈ 𝑥 which is satisfied by elirrv 9595. Usage of this theorem is discouraged because it depends on ax-13 2401. (Contributed by BTernaryTau, 3-Aug-2025.) (New usage is discouraged.)

Assertion

Ref	Expression
axnulg	⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Proof of Theorem axnulg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfnae 2463	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
2		nfcvf 2948	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
3		nfcvd 2923	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧)
4	2, 3	nfeld 2933	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧)
5	4	nfnd 1890	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 ¬ 𝑦 ∈ 𝑧)
6	1, 5	nfald 2358	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦 ¬ 𝑦 ∈ 𝑧)
7		nfvd 1947	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧∀𝑦 ¬ 𝑦 ∈ 𝑥)
8		dveeq2 2407	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥))
9	8	naecoms 2458	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥))
10		elequ2 2159	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
11	10	notbid 321	. . . . 5 ⊢ (𝑧 = 𝑥 → (¬ 𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑥))
12	11	biimpd 232	. . . 4 ⊢ (𝑧 = 𝑥 → (¬ 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑥))
13	12	al2imi 1847	. . 3 ⊢ (∀𝑦 𝑧 = 𝑥 → (∀𝑦 ¬ 𝑦 ∈ 𝑧 → ∀𝑦 ¬ 𝑦 ∈ 𝑥))
14	9, 13	syl6 36	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (∀𝑦 ¬ 𝑦 ∈ 𝑧 → ∀𝑦 ¬ 𝑦 ∈ 𝑥)))
15		elequ1 2151	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
16	15	notbid 321	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥))
17	16	sps 2221	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥))
18	17	dral1 2468	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥))
19	18	biimpd 232	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ¬ 𝑦 ∈ 𝑥))
20		ax-nul 5259	. 2 ⊢ ∃𝑧∀𝑦 ¬ 𝑦 ∈ 𝑧
21		elirrv 9595	. . . 4 ⊢ ¬ 𝑥 ∈ 𝑥
22	21	ax-gen 1827	. . 3 ⊢ ∀𝑥 ¬ 𝑥 ∈ 𝑥
23	22	exgen 2006	. 2 ⊢ ∃𝑥∀𝑥 ¬ 𝑥 ∈ 𝑥
24	6, 7, 14, 19, 20, 23	dvelimexcasei 35755	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-reg 9590

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-cleq 2752  df-clel 2835  df-nfc 2909

This theorem is used by: (None)