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Metamath Proof Explorer

Theorem 0sdom

Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004)

Ref Expression
Hypothesis 0sdom.1 ⊢ A ∈ V
Assertion 0sdom ⊢ ∅ ≺ A ↔ A ≠ ∅

Proof

Step Hyp Ref Expression
1 0sdom.1 ⊢ A ∈ V
2 0sdomg ⊢ A ∈ V → ∅ ≺ A ↔ A ≠ ∅
3 1 2 ax-mp ⊢ ∅ ≺ A ↔ A ≠ ∅