Theorem vn0 4299

Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2116, df-clel 2812. (Revised by GG, 6-Sep-2024.)

Assertion

Ref	Expression
vn0	⊢ V ≠ ∅

Proof of Theorem vn0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vextru 2722	. . . . . . 7 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
2		fal 1556	. . . . . . 7 ⊢ ¬ ⊥
3	1, 2	2th 264	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ¬ ⊥)
4		xor3 382	. . . . . 6 ⊢ (¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) ↔ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ¬ ⊥))
5	3, 4	mpbir 231	. . . . 5 ⊢ ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
6	5	exgen 1976	. . . 4 ⊢ ∃𝑦 ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
7		exnal 1829	. . . 4 ⊢ (∃𝑦 ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) ↔ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
8	6, 7	mpbi 230	. . 3 ⊢ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
9		dfv2 3445	. . . . 5 ⊢ V = {𝑥 ∣ ⊤}
10		dfnul4 4289	. . . . 5 ⊢ ∅ = {𝑥 ∣ ⊥}
11	9, 10	eqeq12i 2755	. . . 4 ⊢ (V = ∅ ↔ {𝑥 ∣ ⊤} = {𝑥 ∣ ⊥})
12		biidd 262	. . . . 5 ⊢ (𝑥 = 𝑦 → (⊥ ↔ ⊥))
13	12	eqabbw 2810	. . . 4 ⊢ ({𝑥 ∣ ⊤} = {𝑥 ∣ ⊥} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
14	11, 13	bitri 275	. . 3 ⊢ (V = ∅ ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
15	8, 14	mtbir 323	. 2 ⊢ ¬ V = ∅
16	15	neir 2936	1 ⊢ V ≠ ∅

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1540   = wceq 1542  ⊤wtru 1543  ⊥wfal 1554  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  Vcvv 3442  ∅c0 4287

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-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-ne 2934  df-v 3444  df-dif 3906  df-nul 4288

This theorem is referenced by:  uniintsn  4942  relrelss  6239  imasaddfnlem  17461  imasvscafn  17470  cmpfi  23364  fclscmp  23986  zarcmplem  34059  compne  44796