Theorem old0

Description: No surreal is older than (/) . (Contributed by Scott Fenton, 7-Aug-2024)

	Ref	Expression
Assertion	old0	$$⊢\mathrm{Old}\left(\varnothing \right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		0elon	$$⊢\varnothing \in \mathrm{On}$$
2		oldval	$$⊢\varnothing \in \mathrm{On}\to \mathrm{Old}\left(\varnothing \right)=\bigcup M\left[\varnothing \right]$$
3	1 2	ax-mp	$$⊢\mathrm{Old}\left(\varnothing \right)=\bigcup M\left[\varnothing \right]$$
4		ima0	$$⊢M\left[\varnothing \right]=\varnothing$$
5	4	unieqi	$$⊢\bigcup M\left[\varnothing \right]=\bigcup \varnothing$$
6		uni0	$$⊢\bigcup \varnothing =\varnothing$$
7	3 5 6	3eqtri	$$⊢\mathrm{Old}\left(\varnothing \right)=\varnothing$$