Theorem vn0 4342

Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2100, df-clel 2806. (Revised by Gino Giotto, 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		fal 1547	. . . . . . 7 ⊢ ¬ ⊥
2		pm5.501 365	. . . . . . . 8 ⊢ (⊤ → (⊥ ↔ (⊤ ↔ ⊥)))
3	2	mptru 1540	. . . . . . 7 ⊢ (⊥ ↔ (⊤ ↔ ⊥))
4	1, 3	mtbi 321	. . . . . 6 ⊢ ¬ (⊤ ↔ ⊥)
5	4	exgen 1970	. . . . 5 ⊢ ∃𝑦 ¬ (⊤ ↔ ⊥)
6		exnal 1821	. . . . 5 ⊢ (∃𝑦 ¬ (⊤ ↔ ⊥) ↔ ¬ ∀𝑦(⊤ ↔ ⊥))
7	5, 6	mpbi 229	. . . 4 ⊢ ¬ ∀𝑦(⊤ ↔ ⊥)
8		df-clab 2706	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ [𝑦 / 𝑥]⊤)
9		sbv 2083	. . . . . . 7 ⊢ ([𝑦 / 𝑥]⊤ ↔ ⊤)
10	8, 9	bitr2i 275	. . . . . 6 ⊢ (⊤ ↔ 𝑦 ∈ {𝑥 ∣ ⊤})
11		df-clab 2706	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∣ ⊥} ↔ [𝑦 / 𝑥]⊥)
12		sbv 2083	. . . . . . 7 ⊢ ([𝑦 / 𝑥]⊥ ↔ ⊥)
13	11, 12	bitr2i 275	. . . . . 6 ⊢ (⊥ ↔ 𝑦 ∈ {𝑥 ∣ ⊥})
14	10, 13	bibi12i 338	. . . . 5 ⊢ ((⊤ ↔ ⊥) ↔ (𝑦 ∈ {𝑥 ∣ ⊤} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}))
15	14	albii 1813	. . . 4 ⊢ (∀𝑦(⊤ ↔ ⊥) ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}))
16	7, 15	mtbi 321	. . 3 ⊢ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ 𝑦 ∈ {𝑥 ∣ ⊥})
17		dfcleq 2721	. . . 4 ⊢ ({𝑥 ∣ ⊤} = {𝑥 ∣ ⊥} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}))
18		dfv2 3476	. . . . . 6 ⊢ V = {𝑥 ∣ ⊤}
19	18	eqcomi 2737	. . . . 5 ⊢ {𝑥 ∣ ⊤} = V
20		dfnul4 4328	. . . . . 6 ⊢ ∅ = {𝑥 ∣ ⊥}
21	20	eqcomi 2737	. . . . 5 ⊢ {𝑥 ∣ ⊥} = ∅
22	19, 21	eqeq12i 2746	. . . 4 ⊢ ({𝑥 ∣ ⊤} = {𝑥 ∣ ⊥} ↔ V = ∅)
23	17, 22	bitr3i 276	. . 3 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ 𝑦 ∈ {𝑥 ∣ ⊥}) ↔ V = ∅)
24	16, 23	mtbi 321	. 2 ⊢ ¬ V = ∅
25	24	neir 2940	1 ⊢ V ≠ ∅

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1531   = wceq 1533  ⊤wtru 1534  ⊥wfal 1545  ∃wex 1773  [wsb 2059   ∈ wcel 2098  {cab 2705   ≠ wne 2937  Vcvv 3473  ∅c0 4326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-ne 2938  df-v 3475  df-dif 3952  df-nul 4327

This theorem is referenced by:  uniintsn  4994  relrelss  6282  imasaddfnlem  17517  imasvscafn  17526  cmpfi  23332  fclscmp  23954  zarcmplem  33515  compne  43909