Theorem noelOLD 4287

Description: Obsolete version of noel 4286 as of 11-Oct-2026. (Contributed by NM, 21-Jun-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
noelOLD	⊢ ¬ 𝐴 ∈ ∅

Proof of Theorem noelOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nsb 2145	. . . . . 6 ⊢ (∀𝑦 ¬ ⊥ → ¬ [𝑥 / 𝑦]⊥)
2		fal 1584	. . . . . 6 ⊢ ¬ ⊥
3	1, 2	mpg 1830	. . . . 5 ⊢ ¬ [𝑥 / 𝑦]⊥
4		dfnul4 4283	. . . . . . 7 ⊢ ∅ = {𝑦 ∣ ⊥}
5	4	eleq2i 2856	. . . . . 6 ⊢ (𝑥 ∈ ∅ ↔ 𝑥 ∈ {𝑦 ∣ ⊥})
6		df-clab 2742	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
7	5, 6	bitri 278	. . . . 5 ⊢ (𝑥 ∈ ∅ ↔ [𝑥 / 𝑦]⊥)
8	3, 7	mtbir 326	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
9	8	intnan 492	. . 3 ⊢ ¬ (𝑥 = 𝐴 ∧ 𝑥 ∈ ∅)
10	9	nex 1833	. 2 ⊢ ¬ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ ∅)
11		dfclel 2840	. 2 ⊢ (𝐴 ∈ ∅ ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ ∅))
12	10, 11	mtbir 326	1 ⊢ ¬ 𝐴 ∈ ∅

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ⊥wfal 1582  ∃wex 1812  [wsb 2101   ∈ wcel 2147  {cab 2741  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-dif 4050  df-nul 4282

This theorem is used by: (None)