Theorem dffin2 ≪ | index | src | ≫

theorem dffin2 (A: set) {n: nat}: $ finite A <-> E. n A C_ upto n $;

Step	Hyp	Ref	Expression
1		bicom	(x e. upto n <-> x < n) -> (x < n <-> x e. upto n)
2		elupto	x e. upto n <-> x < n
3	1, 2	ax_mp	x < n <-> x e. upto n
4	3	imeq2i	x e. A -> x < n <-> x e. A -> x e. upto n
5	4	aleqi	A. x (x e. A -> x < n) <-> A. x (x e. A -> x e. upto n)
6	5	conv subset	A. x (x e. A -> x < n) <-> A C_ upto n
7	6	exeqi	E. n A. x (x e. A -> x < n) <-> E. n A C_ upto n
8	7	conv finite	finite A <-> E. n A C_ upto 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_the (theid, the0), axs_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)