The real existence of self is Atma (also called as atman). It is the underlying metaphysical self. In simple words, 'I exist', this infinite consciousness which is not mind or body is Atma. It is one and same with the infinite Brahman. According to advaita doctrine, it is because of Maya (illusion) that we feel our body alone as self and not real self which is Atman.

The Isovasya Upanishad mantra illustrate the nature of Atman and its relationship with Brahman.

Purnamidam Purnamada: Purnat Purnamudachyate

Purnasya Purnamadaya Purnamevaavasishyate

Atma as Infinite (poorna) when combine with Brahman, another infinite we get same Brahman, the infinite as a result. Similarly when an Atman as infinite is deducted from infinite (Brahman), infinite (Atman) remains.

The meaning of the mantra is better explained by sage Ramakrishna by a synonym that Atma in an individual is like a drop of water in sea and God as the whole water of the sea. Each drop of water is sea just like the whole sea. But, the drop loose the magnanimity when it is not in touch with the whole. The drop of water can restore its magnanimity when it is combined with the whole sea.