Theorem leftirr

Description: No surreal is a member of its left set. (Contributed by Scott Fenton, 20-Aug-2024)

	Ref	Expression
Assertion	leftirr	$$⊢X\in \mathrm{No}\to \neg X\in L\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		oldirr	$$⊢\neg X\in \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)$$
2		leftssold	$$⊢X\in \mathrm{No}\to L\left(X\right)\subseteq \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)$$
3	2	sseld	$$⊢X\in \mathrm{No}\to \left(X\in L\left(X\right)\to X\in \mathrm{Old}\left(\mathrm{bday}\left(X\right)\right)\right)$$
4	1 3	mtoi	$$⊢X\in \mathrm{No}\to \neg X\in L\left(X\right)$$