Theorem 0sdom1dom

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

	Ref	Expression
Assertion	0sdom1dom	$$⊢\varnothing \prec A\leftrightarrow 1_{𝑜}\preccurlyeq A$$

Proof

Step	Hyp	Ref	Expression
1		peano1	$$⊢\varnothing \in \mathrm{\omega }$$
2		sucdom	$$⊢\varnothing \in \mathrm{\omega }\to \left(\varnothing \prec A\leftrightarrow \mathrm{suc}\varnothing \preccurlyeq A\right)$$
3	1 2	ax-mp	$$⊢\varnothing \prec A\leftrightarrow \mathrm{suc}\varnothing \preccurlyeq A$$
4		df-1o	$$⊢1_{𝑜}=\mathrm{suc}\varnothing$$
5	4	breq1i	$$⊢1_{𝑜}\preccurlyeq A\leftrightarrow \mathrm{suc}\varnothing \preccurlyeq A$$
6	3 5	bitr4i	$$⊢\varnothing \prec A\leftrightarrow 1_{𝑜}\preccurlyeq A$$