The Class of Series – Parallel Graphs. Part I

We adopt the following rules: n is a natural number and x1, x2, x3, x4, x5, y1, y2, y3 are sets. Let x1, x2, x3, x4, x5 be sets. We say that x1, x2, x3, x4, x5 are mutually different if and only if: (Def. 1) x1 6= x2 and x1 6= x3 and x1 6= x4 and x1 6= x5 and x2 6= x3 and x2 6= x4 and x2 6= x5 and x3 6= x4 and x3 6= x5 and x4 6= x5. Next we state several propositions: (1) If x1, x2, x3, x4, x5 are mutually different, then card{x1, x2, x3, x4, x5} = 5. (2) 4 = {0, 1, 2, 3}. (3) [: {x1, x2, x3}, {y1, y2, y3} :] = {〈x1, y1〉, 〈x1, y2〉, 〈x1, y3〉, 〈x2, y1〉, 〈x2, y2〉, 〈x2, y3〉, 〈x3, y1〉, 〈x3, y2〉, 〈x3, y3〉}. (4) For every set x and for every natural number n such that x ∈ n holds x is a natural number.