Theorem leftirr

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

	Ref	Expression
Assertion	leftirr	⊢ ( 𝑋 ∈ No → ¬ 𝑋 ∈ ( L ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		oldirr	⊢ ¬ 𝑋 ∈ ( O ‘ ( bday ‘ 𝑋 ) )
2		leftssold	⊢ ( 𝑋 ∈ No → ( L ‘ 𝑋 ) ⊆ ( O ‘ ( bday ‘ 𝑋 ) ) )
3	2	sseld	⊢ ( 𝑋 ∈ No → ( 𝑋 ∈ ( L ‘ 𝑋 ) → 𝑋 ∈ ( O ‘ ( bday ‘ 𝑋 ) ) ) )
4	1 3	mtoi	⊢ ( 𝑋 ∈ No → ¬ 𝑋 ∈ ( L ‘ 𝑋 ) )