Nylon was made in 1931 and Rayon was made in 1880. So, how was nylon is the first synthetic fibre?

Rayon, invented in 1846, began to be manufactured in the United States in 1911. Called artificial silk until 1924 when the name rayon was coined, rayon was a less expensive alternative to silk clothing and accessories. 

Rayon is the first man-made fibre. It is obtained by chemical treatment of wood pulp. So it is also known as a semi-synthetic fibre.

The first sample of Nylon was produced on February 28, 1935, at DuPont's research facility at the DuPont Experimental Station. It had all the desired properties of elasticity and strength.

Nylon, the first synthetic fibre in the "fully synthetic" sense.