Theorem axnulg 35333

Description: A generalization of ax-nul 5235 in which 𝑥 and 𝑦 need not be distinct. This theorem scheme bundles ax-nul 5235 with the degenerate instance ∃𝑥∀𝑥¬ 𝑥 ∈ 𝑥 which is satisfied by elirrv 9509. Usage of this theorem is discouraged because it depends on ax-13 2380. (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 2442	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
2		nfcvf 2928	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
3		nfcvd 2903	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧)
4	2, 3	nfeld 2913	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧)
5	4	nfnd 1865	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 ¬ 𝑦 ∈ 𝑧)
6	1, 5	nfald 2337	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦 ¬ 𝑦 ∈ 𝑧)
7		nfvd 1922	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧∀𝑦 ¬ 𝑦 ∈ 𝑥)
8		dveeq2 2386	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥))
9	8	naecoms 2437	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥))
10		elequ2 2134	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
11	10	notbid 319	. . . . 5 ⊢ (𝑧 = 𝑥 → (¬ 𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑥))
12	11	biimpd 230	. . . 4 ⊢ (𝑧 = 𝑥 → (¬ 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑥))
13	12	al2imi 1822	. . 3 ⊢ (∀𝑦 𝑧 = 𝑥 → (∀𝑦 ¬ 𝑦 ∈ 𝑧 → ∀𝑦 ¬ 𝑦 ∈ 𝑥))
14	9, 13	syl6 35	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → (∀𝑦 ¬ 𝑦 ∈ 𝑧 → ∀𝑦 ¬ 𝑦 ∈ 𝑥)))
15		elequ1 2126	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
16	15	notbid 319	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥))
17	16	sps 2197	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥))
18	17	dral1 2447	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥))
19	18	biimpd 230	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ¬ 𝑦 ∈ 𝑥))
20		ax-nul 5235	. 2 ⊢ ∃𝑧∀𝑦 ¬ 𝑦 ∈ 𝑧
21		elirrv 9509	. . . 4 ⊢ ¬ 𝑥 ∈ 𝑥
22	21	ax-gen 1802	. . 3 ⊢ ∀𝑥 ¬ 𝑥 ∈ 𝑥
23	22	exgen 1981	. 2 ⊢ ∃𝑥∀𝑥 ¬ 𝑥 ∈ 𝑥
24	6, 7, 14, 19, 20, 23	dvelimexcasei 35267	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-13 2380  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-reg 9504

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-nf 1791  df-cleq 2732  df-clel 2815  df-nfc 2889

This theorem is referenced by: (None)