It is narrated from Umm al-Mu'mineen Sayyida Aisha (may Allah be pleased with her) that she borrowed a necklace from Sayyida Asma (may Allah be pleased with her) and it was lost. So the Messenger of Allah (peace and blessings be upon him) sent some of his companions (may Allah be pleased with them) to search for it. The time for prayer came and no water was found, so those people performed the prayer without ablution. When they returned to the Messenger of Allah (peace and blessings be upon him), they mentioned this incident. At that very moment, the verse of tayammum was revealed. Then Sayyiduna Usayd bin Hudayr (may Allah be pleased with him) said: May Allah reward you with goodness. By Allah, whenever any difficulty has come upon you, Allah has removed it and made it a source of blessing for the Muslims.
Hadith Takhrijیہ حدیث صحیح متفق علیہ ہے۔ دیکھئے: [بخاري 334] ، [مسلم 367] و [صحيح ابن حبان 1300]
Brief Explanation
(Commentary on Hadiths 767 to 769)
From this it is understood that if neither earth (dust) nor water is available, then the prayer should still be performed.
Imam Shawkani rahimahullah has written in [نيل الأوطار 267/1] under the discussion of this hadith: The scholars of investigation have deduced from this hadith that if someone cannot find either water or earth (dust), even then prayer remains obligatory. In the hadith, those people who are mentioned did not find water, yet, considering the prayer obligatory, they performed it. If it were not correct for them to pray without ablution (wudu), then the Messenger of Allah sallallahu alayhi wa sallam would certainly have objected to them.
Therefore, this is the same ruling for one who finds neither water nor earth (dust), because purification is attained through these two things, so it will be necessary for him to perform the prayer. This is the fatwa of the majority of the hadith scholars.