Theorem left0s

Description: The left set of 0s is empty. (Contributed by Scott Fenton, 20-Aug-2024)

	Ref	Expression
Assertion	left0s	⊢ ( L ‘ 0s ) = ∅

Proof

Step	Hyp	Ref	Expression
1		leftssold	⊢ ( L ‘ 0s ) ⊆ ( O ‘ ( bday ‘ 0s ) )
2		bday0s	⊢ ( bday ‘ 0s ) = ∅
3	2	fveq2i	⊢ ( O ‘ ( bday ‘ 0s ) ) = ( O ‘ ∅ )
4		old0	⊢ ( O ‘ ∅ ) = ∅
5	3 4	eqtri	⊢ ( O ‘ ( bday ‘ 0s ) ) = ∅
6		sseq0	⊢ ( ( ( L ‘ 0s ) ⊆ ( O ‘ ( bday ‘ 0s ) ) ∧ ( O ‘ ( bday ‘ 0s ) ) = ∅ ) → ( L ‘ 0s ) = ∅ )
7	1 5 6	mp2an	⊢ ( L ‘ 0s ) = ∅