# Math Help - underlying set

1. ## underlying set

let Phi_n denote the sentence for all x_1 for all x_2 ...for all x_{n-1} there exists x_n (neg (x_{n} =x_1) ) /\ neg (x_{n} =x_2) /\.../\
neg(x_{n} = x_{n-1} ).
how can I show that every model of Phi_n must have at least n elements in the underlying set .

2. If a model has fewer than n elements, then you can instantiate x_1, ..., x_{n-1} with all of these elements (possibly, with repetition). Then you can't find x_n that is different from all of x_1, ..., x_{n-1}.

3. Thanks alot emakarov