Theorem noelOLD 4265

Description: Obsolete version of noel 4264 as of 18-Sep-2024. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
noelOLD	⊢ ¬ 𝐴 ∈ ∅

Proof of Theorem noelOLD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm3.24 403	. . . . . . 7 ⊢ ¬ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)
2	1	nex 1803	. . . . . 6 ⊢ ¬ ∃𝑦(𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)
3		df-clab 2716	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)} ↔ [𝑥 / 𝑦](𝑦 ∈ V ∧ ¬ 𝑦 ∈ V))
4		spsbe 2085	. . . . . . 7 ⊢ ([𝑥 / 𝑦](𝑦 ∈ V ∧ ¬ 𝑦 ∈ V) → ∃𝑦(𝑦 ∈ V ∧ ¬ 𝑦 ∈ V))
5	3, 4	sylbi 216	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)} → ∃𝑦(𝑦 ∈ V ∧ ¬ 𝑦 ∈ V))
6	2, 5	mto 196	. . . . 5 ⊢ ¬ 𝑥 ∈ {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)}
7		df-nul 4257	. . . . . . 7 ⊢ ∅ = (V ∖ V)
8		df-dif 3890	. . . . . . 7 ⊢ (V ∖ V) = {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)}
9	7, 8	eqtri 2766	. . . . . 6 ⊢ ∅ = {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)}
10	9	eleq2i 2830	. . . . 5 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑦 ∣ (𝑦 ∈ V ∧ ¬ 𝑦 ∈ V)})
11	6, 10	mtbir 323	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
12	11	intnan 487	. . 3 ⊢ ¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ ∅)
13	12	nex 1803	. 2 ⊢ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ ∅)
14		dfclel 2817	. 2 ⊢ (𝐴 ∈ ∅ ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ ∅))
15	13, 14	mtbir 323	1 ⊢ ¬ 𝐴 ∈ ∅

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1539  ∃wex 1782  [wsb 2067   ∈ wcel 2106  {cab 2715  Vcvv 3432   ∖ cdif 3884  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-dif 3890  df-nul 4257

This theorem is referenced by: (None)