Theorem vn0 4298

Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2145, df-clel 2838. (Revised by GG, 6-Sep-2024.) (Proof shortened by BJ, 12-Jul-2026.)

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 1584	. . . 4 ⊢ ¬ ⊥
2		vextru 2748	. . . . . 6 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
3		biimp 218	. . . . . 6 ⊢ ((𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) → (𝑦 ∈ {𝑥 ∣ ⊤} → ⊥))
4	2, 3	mpi 21	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) → ⊥)
5	4	spsv 2017	. . . 4 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) → ⊥)
6	1, 5	mto 200	. . 3 ⊢ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
7		dfv2 3458	. . . . 5 ⊢ V = {𝑥 ∣ ⊤}
8		dfnul4 4288	. . . . 5 ⊢ ∅ = {𝑥 ∣ ⊥}
9	7, 8	eqeq12i 2781	. . . 4 ⊢ (V = ∅ ↔ {𝑥 ∣ ⊤} = {𝑥 ∣ ⊥})
10		biidd 265	. . . . 5 ⊢ (𝑥 = 𝑦 → (⊥ ↔ ⊥))
11	10	eqabbw 2836	. . . 4 ⊢ ({𝑥 ∣ ⊤} = {𝑥 ∣ ⊥} ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
12	9, 11	bitri 278	. . 3 ⊢ (V = ∅ ↔ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥))
13	6, 12	mtbir 326	. 2 ⊢ ¬ V = ∅
14	13	neir 2961	1 ⊢ V ≠ ∅

Syntax hints:   ↔ wb 209  ∀wal 1568   = wceq 1570  ⊤wtru 1571  ⊥wfal 1582   ∈ wcel 2143  {cab 2741   ≠ wne 2958  Vcvv 3455  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-ne 2959  df-v 3457  df-dif 3908  df-nul 4287

This theorem is referenced by:  uniintsn  4950  relrelss  6274  imasaddfnlem  17577  imasvscafn  17586  cmpfi  23565  fclscmp  24187  zarcmplem  34271  compne  45170