Write the smallest 4-digit number of four different digits.

The digits available are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Thousands digit: The smallest digit is 0, but a number cannot start with 0 if it's meant to be a 4-digit number. So, the smallest non-zero digit, which is 1, must be in the thousands place.

Current digits used: {1}

Hundreds digit: Now that 1 is used, the smallest remaining digit available is 0.

Current digits used: {1, 0}

Tens digit: With 1 and 0 used, the next smallest digit available is 2.

Current digits used: {1, 0, 2}

Units digit: With 1, 0, and 2 used, the next smallest digit available is 3.

Current digits used: {1, 0, 2, 3}

Combining these digits in order, the smallest 4-digit number with four different digits is 1023.

Hence, the smallest 4-digit number with four different digits is 1,023.