How do you show that 1 ml = 1 cc? Why is it true?

Unfortunately, 1ml does not equal 1cc. There is a difference when you go to 5 decimal places.

The metric system has it roots in Paris in 1793. In addition to decimal system proposals, units of length, mass and volume were provisionally created. (A brass standard of the provisional metre was made: it is preserved in the Conservatoire des Arts et Métiers, Paris.) This was when the LITRE was first defined as a measure for liquids, this being an appropriately sized volume for commercial use. As the 19th century drew to a close, very precise measurements were needed in the fields of Physics, Chemistry and engineering. In 1889 the "Standard Kilogramme" was created. This was supposed to be the same as 1 litre of distilled water at its maximum density -ie. at a temperature of 4 degrees celsius. The LITRE then became officially defined as 1 kilogramme of pure water at 4°C. Unfortunately there was a very small error and it was not until 1907 that it was detected. The "1889 Standard kilogramme" was discovered to have a mass of 1000.028 cc of pure water at 4°C and so it followed that a LITRE was 1000.028 cc. A decision was taken to leave the kilogramme as the "Standard" but to divide the LITRE into 1000 equal parts and to call this division by a new name the millilitre (ml).
From 1907 millilitres were used as the standard unit of liquid and volume measurement.
So 1 millilitre then equalled 1.000028 cc and 1 cc equalled 0.999972 ml.
Accordingly 1 millilitre of pure water at 4°C had a mass of 1 gramme

In 1964 the General Conference of Weights and Measurements re-defined the LITRE as a true measurement of volume and so equal to 1000 centimetres cubed (cm3 or cc). This changed the specification of the litre, by the fact that its new definition is directly related to the metre as a measurement of volume and no longer to the kilogramme.
However, the millilitre remained as per the original specification and the ml calibration of scientific vessels used in very accurate analytical work is not 1/1000th of a litre where a litre = 1000cc, but where 1ml of pure water at 4°C has a mass of 1 gramme.
So 1ml does not equal 1cc