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Metamath Proof Explorer

Theorem un0

Description: The union of a class with the empty set is itself. Theorem 24 of Suppes p. 27. (Contributed by NM, 15-Jul-1993)

Ref Expression
Assertion un0 ⊢ A ∪ ∅ = A

Proof

Step Hyp Ref Expression
1 noel ⊢ ¬ x ∈ ∅
2 1 biorfri ⊢ x ∈ A ↔ x ∈ A ∨ x ∈ ∅
3 2 bicomi ⊢ x ∈ A ∨ x ∈ ∅ ↔ x ∈ A
4 3 uneqri ⊢ A ∪ ∅ = A