Theorem dom0

Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004)

	Ref	Expression
Assertion	dom0	⊢ ( 𝐴 ≼ ∅ ↔ 𝐴 = ∅ )

Proof

Step	Hyp	Ref	Expression
1		reldom	⊢ Rel ≼
2	1	brrelex1i	⊢ ( 𝐴 ≼ ∅ → 𝐴 ∈ V )
3		0domg	⊢ ( 𝐴 ∈ V → ∅ ≼ 𝐴 )
4	2 3	syl	⊢ ( 𝐴 ≼ ∅ → ∅ ≼ 𝐴 )
5	4	pm4.71i	⊢ ( 𝐴 ≼ ∅ ↔ ( 𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴 ) )
6		sbthb	⊢ ( ( 𝐴 ≼ ∅ ∧ ∅ ≼ 𝐴 ) ↔ 𝐴 ≈ ∅ )
7		en0	⊢ ( 𝐴 ≈ ∅ ↔ 𝐴 = ∅ )
8	5 6 7	3bitri	⊢ ( 𝐴 ≼ ∅ ↔ 𝐴 = ∅ )