Theorem snfin ≪ | index | src | ≫

theorem snfin (n: nat) {x: nat}: $ finite {x | x = n} $;

Step	Hyp	Ref	Expression
1		finss	{x \| x = n} C_ {x \| x <= n} -> finite {x \| x <= n} -> finite {x \| x = n}
2		ssab	A. x (x = n -> x <= n) <-> {x \| x = n} C_ {x \| x <= n}
3		eqle	x = n -> x <= n
4	3	ax_gen	A. x (x = n -> x <= n)
5	2, 4	mpbi	{x \| x = n} C_ {x \| x <= n}
6	1, 5	ax_mp	finite {x \| x <= n} -> finite {x \| x = n}
7		lefin	finite {x \| x <= n}
8	6, 7	ax_mp	finite {x \| x = n}

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8), axs_peano (peano2, peano5, addeq, add0, addS)