Theorem right0s

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

	Ref	Expression
Assertion	right0s	⊢ ( R ‘ 0s ) = ∅

Proof

Step	Hyp	Ref	Expression
1		0sno	⊢ 0s ∈ No
2		rightssold	⊢ ( 0s ∈ No → ( R ‘ 0s ) ⊆ ( O ‘ ( bday ‘ 0s ) ) )
3	1 2	ax-mp	⊢ ( R ‘ 0s ) ⊆ ( O ‘ ( bday ‘ 0s ) )
4		bday0s	⊢ ( bday ‘ 0s ) = ∅
5	4	fveq2i	⊢ ( O ‘ ( bday ‘ 0s ) ) = ( O ‘ ∅ )
6		old0	⊢ ( O ‘ ∅ ) = ∅
7	5 6	eqtri	⊢ ( O ‘ ( bday ‘ 0s ) ) = ∅
8		sseq0	⊢ ( ( ( R ‘ 0s ) ⊆ ( O ‘ ( bday ‘ 0s ) ) ∧ ( O ‘ ( bday ‘ 0s ) ) = ∅ ) → ( R ‘ 0s ) = ∅ )
9	3 7 8	mp2an	⊢ ( R ‘ 0s ) = ∅