Theorem axnulg 35140

Description: A generalization of ax-nul 5246 in which 𝑥 and 𝑦 need not be distinct. Note that it is possible to use axc7e 2321 to derive elirrv 9490 from this theorem, which justifies the dependency on ax-reg 9485. Usage of this theorem is discouraged because it depends on ax-13 2374. (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 2436	. . 3 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
2		nfcvf 2922	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
3		nfcvd 2896	. . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑧)
4	2, 3	nfeld 2907	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝑧)
5	4	nfnd 1859	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 ¬ 𝑦 ∈ 𝑧)
6	1, 5	nfald 2331	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑦 ¬ 𝑦 ∈ 𝑧)
7		nfvd 1916	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧∀𝑦 ¬ 𝑦 ∈ 𝑥)
8		dveeq2 2380	. . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥))
9	8	naecoms 2431	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑥 → ∀𝑦 𝑧 = 𝑥))
10		elequ2 2128	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
11	10	notbid 318	. . . . 5 ⊢ (𝑧 = 𝑥 → (¬ 𝑦 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑥))
12	11	biimpd 229	. . . 4 ⊢ (𝑧 = 𝑥 → (¬ 𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑥))
13	12	al2imi 1816	. . 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 2120	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
16	15	notbid 318	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥))
17	16	sps 2190	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑥))
18	17	dral1 2441	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥))
19	18	biimpd 229	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ¬ 𝑦 ∈ 𝑥))
20		ax-nul 5246	. 2 ⊢ ∃𝑧∀𝑦 ¬ 𝑦 ∈ 𝑧
21		elirrv 9490	. . . 4 ⊢ ¬ 𝑥 ∈ 𝑥
22	21	ax-gen 1796	. . 3 ⊢ ∀𝑥 ¬ 𝑥 ∈ 𝑥
23	22	exgen 1975	. 2 ⊢ ∃𝑥∀𝑥 ¬ 𝑥 ∈ 𝑥
24	6, 7, 14, 19, 20, 23	dvelimexcasei 35111	1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-13 2374  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-reg 9485

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-cleq 2725  df-clel 2808  df-nfc 2882

This theorem is referenced by: (None)