Theorem vn0 4290

Description: The universal class is not equal to the empty set. (Contributed by NM, 11-Sep-2008.) Avoid ax-8 2146, df-clel 2835. (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 1583	. . . 4 ⊢ ¬ ⊥
2		vextru 2745	. . . . . 6 ⊢ 𝑦 ∈ {𝑥 ∣ ⊤}
3		biimp 218	. . . . . 6 ⊢ ((𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) → (𝑦 ∈ {𝑥 ∣ ⊤} → ⊥))
4	2, 3	mpi 21	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) → ⊥)
5	4	spsv 2019	. . . 4 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥) → ⊥)
6	1, 5	mto 200	. . 3 ⊢ ¬ ∀𝑦(𝑦 ∈ {𝑥 ∣ ⊤} ↔ ⊥)
7		dfv2 3453	. . . . 5 ⊢ V = {𝑥 ∣ ⊤}
8		dfnul4 4279	. . . . 5 ⊢ ∅ = {𝑥 ∣ ⊥}
9	7, 8	eqeq12i 2778	. . . 4 ⊢ (V = ∅ ↔ {𝑥 ∣ ⊤} = {𝑥 ∣ ⊥})
10		biidd 265	. . . . 5 ⊢ (𝑥 = 𝑦 → (⊥ ↔ ⊥))
11	10	eqabbw 2833	. . . 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 2958	1 ⊢ V ≠ ∅

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567   = wceq 1569  ⊤wtru 1570  ⊥wfal 1581   ∈ wcel 2144  {cab 2738   ≠ wne 2955  Vcvv 3450  ∅c0 4277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-ne 2956  df-v 3452  df-dif 4046  df-nul 4278

This theorem is used by:  uniintsn  4944  relrelss  6278  imasaddfnlem  17709  imasvscafn  17718  cmpfi  23735  fclscmp  24358  zarcmplem  34563  compne  45560