Theorem ordne0gt0

Description: Ordinal zero is less than every non-zero ordinal. Theorem 1.10 of Schloeder p. 2. Closely related to ord0eln0 . (Contributed by RP, 16-Jan-2025)

	Ref	Expression
Assertion	ordne0gt0	$$⊢\left(\mathrm{Ord}A\wedge A\ne \varnothing \right)\to \varnothing \in A$$

Proof

Step	Hyp	Ref	Expression
1		ord0eln0	$$⊢\mathrm{Ord}A\to \left(\varnothing \in A\leftrightarrow A\ne \varnothing \right)$$
2	1	biimpar	$$⊢\left(\mathrm{Ord}A\wedge A\ne \varnothing \right)\to \varnothing \in A$$