Theorem left0s

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

	Ref	Expression
Assertion	left0s	$$⊢L\left(0_s\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		leftssold	$$⊢L\left(0_s\right)\subseteq \mathrm{Old}\left(\mathrm{bday}\left(0_s\right)\right)$$
2		bday0	$$⊢\mathrm{bday}\left(0_s\right)=\varnothing$$
3	2	fveq2i	$$⊢\mathrm{Old}\left(\mathrm{bday}\left(0_s\right)\right)=\mathrm{Old}\left(\varnothing \right)$$
4		old0	$$⊢\mathrm{Old}\left(\varnothing \right)=\varnothing$$
5	3 4	eqtri	$$⊢\mathrm{Old}\left(\mathrm{bday}\left(0_s\right)\right)=\varnothing$$
6		sseq0	$$⊢\left(L\left(0_s\right)\subseteq \mathrm{Old}\left(\mathrm{bday}\left(0_s\right)\right)\wedge \mathrm{Old}\left(\mathrm{bday}\left(0_s\right)\right)=\varnothing \right)\to L\left(0_s\right)=\varnothing$$
7	1 5 6	mp2an	$$⊢L\left(0_s\right)=\varnothing$$