Theorem vn0 4273

Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2121, df-clel 2814. (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 2724	. . . . . . 7 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
2		fal 1561	. . . . . . 7 ⊢ ¬ ⊥
3	1, 2	2th 265	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ¬ ⊥)
4		xor3 383	. . . . . 6 ⊢ (¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) ↔ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ¬ ⊥))
5	3, 4	mpbir 232	. . . . 5 ⊢ ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
6	5	exgen 1981	. . . 4 ⊢ ∃𝑦 ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
7		exnal 1834	. . . 4 ⊢ (∃𝑦 ¬ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) ↔ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
8	6, 7	mpbi 231	. . 3 ⊢ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
9		dfv2 3434	. . . . 5 ⊢ V = {𝑥 ∣ ⊤}
10		dfnul4 4263	. . . . 5 ⊢ ∅ = {𝑥 ∣ ⊥}
11	9, 10	eqeq12i 2757	. . . 4 ⊢ (V = ∅ ↔ {𝑥 ∣ ⊤} = {𝑥 ∣ ⊥})
12		biidd 263	. . . . 5 ⊢ (𝑥 = 𝑦 → (⊥ ↔ ⊥))
13	12	eqabbw 2812	. . . 4 ⊢ ({𝑥 ∣ ⊤} = {𝑥 ∣ ⊥} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
14	11, 13	bitri 276	. . 3 ⊢ (V = ∅ ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
15	8, 14	mtbir 324	. 2 ⊢ ¬ V = ∅
16	15	neir 2937	1 ⊢ V ≠ ∅

Syntax hints:  ¬ wn 3   ↔ wb 207  ∀wal 1545   = wceq 1547  ⊤wtru 1548  ⊥wfal 1559  ∃wex 1786   ∈ wcel 2119  {cab 2717   ≠ wne 2934  Vcvv 3431  ∅c0 4261

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-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-ne 2935  df-v 3433  df-dif 3886  df-nul 4262

This theorem is referenced by:  uniintsn  4915  relrelss  6224  imasaddfnlem  17483  imasvscafn  17492  cmpfi  23391  fclscmp  24013  zarcmplem  34065  compne  44884