Theorem leftirr

Description: No surreal is a member of its left set. (Contributed by Scott Fenton, 9-Oct-2024)

	Ref	Expression
Assertion	leftirr	⊢ ¬ 𝑋 ∈ ( L ‘ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		oldirr	⊢ ¬ 𝑋 ∈ ( O ‘ ( bday ‘ 𝑋 ) )
2		leftssold	⊢ ( L ‘ 𝑋 ) ⊆ ( O ‘ ( bday ‘ 𝑋 ) )
3	2	sseli	⊢ ( 𝑋 ∈ ( L ‘ 𝑋 ) → 𝑋 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) )
4	1 3	mto	⊢ ¬ 𝑋 ∈ ( L ‘ 𝑋 )