Theorem sdom0

Description: The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003)

	Ref	Expression
Assertion	sdom0	⊢ ¬ 𝐴 ≺ ∅

Proof

Step	Hyp	Ref	Expression
1		relsdom	⊢ Rel ≺
2	1	brrelex1i	⊢ ( 𝐴 ≺ ∅ → 𝐴 ∈ V )
3		0domg	⊢ ( 𝐴 ∈ V → ∅ ≼ 𝐴 )
4	2 3	syl	⊢ ( 𝐴 ≺ ∅ → ∅ ≼ 𝐴 )
5		domnsym	⊢ ( ∅ ≼ 𝐴 → ¬ 𝐴 ≺ ∅ )
6	5	con2i	⊢ ( 𝐴 ≺ ∅ → ¬ ∅ ≼ 𝐴 )
7	4 6	pm2.65i	⊢ ¬ 𝐴 ≺ ∅