Theorem vn0 4297

Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2144, df-clel 2837. (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 2747	. . . . . . 7 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
2		fal 1574	. . . . . . 7 ⊢ ¬ ⊥
3	1, 2	2th 266	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ¬ ⊥)
4		xor3 384	. . . . . 6 ⊢ (¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) ↔ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ¬ ⊥))
5	3, 4	mpbir 233	. . . . 5 ⊢ ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
6	5	exgen 1994	. . . 4 ⊢ ∃𝑦 ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
7		exnal 1847	. . . 4 ⊢ (∃𝑦 ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) ↔ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
8	6, 7	mpbi 232	. . 3 ⊢ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
9		dfv2 3457	. . . . 5 ⊢ V = {𝑥 ∣ ⊤}
10		dfnul4 4287	. . . . 5 ⊢ ∅ = {𝑥 ∣ ⊥}
11	9, 10	eqeq12i 2780	. . . 4 ⊢ (V = ∅ ↔ {𝑥 ∣ ⊤} = {𝑥 ∣ ⊥})
12		biidd 264	. . . . 5 ⊢ (𝑥 = 𝑦 → (⊥ ↔ ⊥))
13	12	eqabbw 2835	. . . 4 ⊢ ({𝑥 ∣ ⊤} = {𝑥 ∣ ⊥} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
14	11, 13	bitri 277	. . 3 ⊢ (V = ∅ ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
15	8, 14	mtbir 325	. 2 ⊢ ¬ V = ∅
16	15	neir 2960	1 ⊢ V ≠ ∅

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1558   = wceq 1560  ⊤wtru 1561  ⊥wfal 1572  ∃wex 1799   ∈ wcel 2142  {cab 2740   ≠ wne 2957  Vcvv 3454  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-ne 2958  df-v 3456  df-dif 3907  df-nul 4286

This theorem is referenced by:  uniintsn  4943  relrelss  6260  imasaddfnlem  17558  imasvscafn  17567  cmpfi  23468  fclscmp  24090  zarcmplem  34178  compne  45016