Metamath Proof Explorer

Theorem eq0

Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of Suppes p. 22. (Contributed by NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024) Avoid ax-8 , df-clel . (Revised by Gino Giotto, 6-Sep-2024)

Ref Expression
Assertion eq0 ⊢ A = ∅ ↔ ∀ x ¬ x ∈ A

Proof

Step Hyp Ref Expression
1 dfnul4 ⊢ ∅ = y | ⊥
2 1 eqeq2i ⊢ A = ∅ ↔ A = y | ⊥
3 dfcleq ⊢ A = y | ⊥ ↔ ∀ x x ∈ A ↔ x ∈ y | ⊥
4 df-clab ⊢ x ∈ y | ⊥ ↔ x y ⊥
5 sbv ⊢ x y ⊥ ↔ ⊥
6 4 5 bitri ⊢ x ∈ y | ⊥ ↔ ⊥
7 6 bibi2i ⊢ x ∈ A ↔ x ∈ y | ⊥ ↔ x ∈ A ↔ ⊥
8 nbfal ⊢ ¬ x ∈ A ↔ x ∈ A ↔ ⊥
9 7 8 bitr4i ⊢ x ∈ A ↔ x ∈ y | ⊥ ↔ ¬ x ∈ A
10 9 albii ⊢ ∀ x x ∈ A ↔ x ∈ y | ⊥ ↔ ∀ x ¬ x ∈ A
11 3 10 bitri ⊢ A = y | ⊥ ↔ ∀ x ¬ x ∈ A
12 2 11 bitri ⊢ A = ∅ ↔ ∀ x ¬ x ∈ A