Theorem oldirr

Description: No surreal is a member of its birthday's old set. (Contributed by Scott Fenton, 10-Aug-2024)

	Ref	Expression
Assertion	oldirr	$$⊢\neg X\in \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bdayelon	$$⊢\mathrm{bday}\left(X\right)\in \mathrm{On}$$
2	1	onirri	$$⊢\neg \mathrm{bday}\left(X\right)\in \mathrm{bday}\left(X\right)$$
3		oldbdayim	$$⊢\left(\mathrm{bday}\left(X\right)\in \mathrm{On}\wedge X\in \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)\right)\to \mathrm{bday}\left(X\right)\in \mathrm{bday}\left(X\right)$$
4	1 3	mpan	$$⊢X\in \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)\to \mathrm{bday}\left(X\right)\in \mathrm{bday}\left(X\right)$$
5	2 4	mto	$$⊢\neg X\in \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)$$