Theorem 0sdomg

Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006)

	Ref	Expression
Assertion	0sdomg	⊢ ( 𝐴 ∈ 𝑉 → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		0domg	⊢ ( 𝐴 ∈ 𝑉 → ∅ ≼ 𝐴 )
2		brsdom	⊢ ( ∅ ≺ 𝐴 ↔ ( ∅ ≼ 𝐴 ∧ ¬ ∅ ≈ 𝐴 ) )
3	2	baib	⊢ ( ∅ ≼ 𝐴 → ( ∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴 ) )
4	1 3	syl	⊢ ( 𝐴 ∈ 𝑉 → ( ∅ ≺ 𝐴 ↔ ¬ ∅ ≈ 𝐴 ) )
5		ensymb	⊢ ( ∅ ≈ 𝐴 ↔ 𝐴 ≈ ∅ )
6		en0	⊢ ( 𝐴 ≈ ∅ ↔ 𝐴 = ∅ )
7	5 6	bitri	⊢ ( ∅ ≈ 𝐴 ↔ 𝐴 = ∅ )
8	7	necon3bbii	⊢ ( ¬ ∅ ≈ 𝐴 ↔ 𝐴 ≠ ∅ )
9	4 8	bitrdi	⊢ ( 𝐴 ∈ 𝑉 → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) )