Theorem leftirr

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

	Ref	Expression
Assertion	leftirr	|- ( X e. No -> -. X e. ( _L ` X ) )

Proof

Step	Hyp	Ref	Expression
1		oldirr	|- -. X e. ( _Old ` ( bday ` X ) )
2		leftssold	|- ( X e. No -> ( _L ` X ) C_ ( _Old ` ( bday ` X ) ) )
3	2	sseld	|- ( X e. No -> ( X e. ( _L ` X ) -> X e. ( _Old ` ( bday ` X ) ) ) )
4	1 3	mtoi	|- ( X e. No -> -. X e. ( _L ` X ) )