Theorem noel

Description: The empty set has no elements. Theorem 6.14 of Quine p. 44. (Contributed by NM, 21-Jun-1993) (Proof shortened by Mario Carneiro, 1-Sep-2015) Remove dependency on ax-10 , ax-11 , and ax-12 . (Revised by Steven Nguyen, 3-May-2023) (Proof shortened by BJ, 23-Sep-2024)

	Ref	Expression
Assertion	noel	⊢ ¬ 𝐴 ∈ ∅

Proof

Step	Hyp	Ref	Expression
1		nsb	⊢ ( ∀ 𝑦 ¬ ⊥ → ¬ [ 𝑥 / 𝑦 ] ⊥ )
2		fal	⊢ ¬ ⊥
3	1 2	mpg	⊢ ¬ [ 𝑥 / 𝑦 ] ⊥
4		dfnul4	⊢ ∅ = { 𝑦 ∣ ⊥ }
5	4	eleq2i	⊢ ( 𝑥 ∈ ∅ ↔ 𝑥 ∈ { 𝑦 ∣ ⊥ } )
6		df-clab	⊢ ( 𝑥 ∈ { 𝑦 ∣ ⊥ } ↔ [ 𝑥 / 𝑦 ] ⊥ )
7	5 6	bitri	⊢ ( 𝑥 ∈ ∅ ↔ [ 𝑥 / 𝑦 ] ⊥ )
8	3 7	mtbir	⊢ ¬ 𝑥 ∈ ∅
9	8	intnan	⊢ ¬ ( 𝑥 = 𝐴 ∧ 𝑥 ∈ ∅ )
10	9	nex	⊢ ¬ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ ∅ )
11		dfclel	⊢ ( 𝐴 ∈ ∅ ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ ∅ ) )
12	10 11	mtbir	⊢ ¬ 𝐴 ∈ ∅