Regarding Yom Kippur, the Torah says:
For on this day He shall effect atonement for you to cleanse you. Before Hashem, you shall be cleansed from all your sins. It is a Sabbath of rest for you, and you shall afflict your souls. It is an eternal statute.
כִּי בַיּוֹם הַזֶּה יְכַפֵּר עֲלֵיכֶם לְטַהֵר אֶתְכֶם מִכֹּל חַטֹּאתֵיכֶם לִפְנֵי יְ-ק-וָֹ-ק תִּטְהָרוּ. שַׁבַּת שַׁבָּתוֹן הִיא לָכֶם וְעִנִּיתֶם אֶת נַפְשֹׁתֵיכֶם חֻקַּת עוֹלָם
Vayikra 16:30-31
Regarding Shabbat, the Torah says:
Six days work may be done, but the seventh day is a Sabbath of complete rest, holy to Hashem, whoever performs work on the Sabbath day shall be put to death
שֵׁשֶׁת יָמִים יֵעָשֶׂה מְלָאכָה וּבַיּוֹם הַשְּׁבִיעִי שַׁבַּת שַׁבָּתוֹן קֹדֶשׁ לה כָּל הָעֹשֶׂה מְלָאכָה בְּיוֹם הַשַּׁבָּת מוֹת יוּמָת
Shmot 31:15
Here Shabbat is referred to as being "holy to Hashem" where it would really seem that this refers to
Yom Kippur since
Yom Kippur is a day devoid of the material, and completely dedicated to the holiness of Hashem.
