Theorem un0

Description: The union of a class with the empty set is itself. Dual of inv1 . Commuted form of 0un . 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