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Metamath Proof Explorer

Theorem 0sdomg

Description: A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006) Avoid ax-pow , ax-un . (Revised by BTernaryTau, 29-Nov-2024)

		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		en0r	⊢ ( ∅ ≈ 𝐴 ↔ 𝐴 = ∅ )
6	5	necon3bbii	⊢ ( ¬ ∅ ≈ 𝐴 ↔ 𝐴 ≠ ∅ )
7	4 6	bitrdi	⊢ ( 𝐴 ∈ 𝑉 → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) )