Theorem 0sdom1dom

Description: Strict dominance over zero is the same as dominance over one. (Contributed by NM, 28-Sep-2004)

	Ref	Expression
Assertion	0sdom1dom	⊢ ( ∅ ≺ 𝐴 ↔ 1o ≼ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		peano1	⊢ ∅ ∈ ω
2		sucdom	⊢ ( ∅ ∈ ω → ( ∅ ≺ 𝐴 ↔ suc ∅ ≼ 𝐴 ) )
3	1 2	ax-mp	⊢ ( ∅ ≺ 𝐴 ↔ suc ∅ ≼ 𝐴 )
4		df-1o	⊢ 1o = suc ∅
5	4	breq1i	⊢ ( 1o ≼ 𝐴 ↔ suc ∅ ≼ 𝐴 )
6	3 5	bitr4i	⊢ ( ∅ ≺ 𝐴 ↔ 1o ≼ 𝐴 )