Theorem vn0 4297

Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2115, df-clel 2811. (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 2721	. . . . . . 7 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
2		fal 1555	. . . . . . 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 1975	. . . 4 ⊢ ∃𝑦 ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
7		exnal 1828	. . . 4 ⊢ (∃𝑦 ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) ↔ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
8	6, 7	mpbi 230	. . 3 ⊢ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
9		dfv2 3443	. . . . 5 ⊢ V = {𝑥 ∣ ⊤}
10		dfnul4 4287	. . . . 5 ⊢ ∅ = {𝑥 ∣ ⊥}
11	9, 10	eqeq12i 2754	. . . 4 ⊢ (V = ∅ ↔ {𝑥 ∣ ⊤} = {𝑥 ∣ ⊥})
12		biidd 262	. . . . 5 ⊢ (𝑥 = 𝑦 → (⊥ ↔ ⊥))
13	12	eqabbw 2809	. . . 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 2935	1 ⊢ V ≠ ∅

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1539   = wceq 1541  ⊤wtru 1542  ⊥wfal 1553  ∃wex 1780   ∈ wcel 2113  {cab 2714   ≠ wne 2932  Vcvv 3440  ∅c0 4285

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-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-ne 2933  df-v 3442  df-dif 3904  df-nul 4286

This theorem is referenced by:  uniintsn  4940  relrelss  6231  imasaddfnlem  17449  imasvscafn  17458  cmpfi  23352  fclscmp  23974  zarcmplem  34038  compne  44691