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Metamath Proof Explorer

Theorem eq0ALT

Description: Alternate proof of eq0 . Shorter, but requiring df-clel , ax-8 . (Contributed by NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by GG and Steven Nguyen, 28-Jun-2024) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion eq0ALT ⊢ A = ∅ ↔ ∀ x ¬ x ∈ A

Proof

Step Hyp Ref Expression
1 dfcleq ⊢ A = ∅ ↔ ∀ x x ∈ A ↔ x ∈ ∅
2 noel ⊢ ¬ x ∈ ∅
3 2 nbn ⊢ ¬ x ∈ A ↔ x ∈ A ↔ x ∈ ∅
4 3 albii ⊢ ∀ x ¬ x ∈ A ↔ ∀ x x ∈ A ↔ x ∈ ∅
5 1 4 bitr4i ⊢ A = ∅ ↔ ∀ x ¬ x ∈ A