Theorem all0 ≪ | index | src | ≫

theorem all0 (A: set): $ all A 0 $;

Step	Hyp	Ref	Expression
1		absurd	~x e. lmems 0 -> x e. lmems 0 -> x e. A
2		lmem0	~x IN 0
3	2	conv lmem	~x e. lmems 0
4	1, 3	ax_mp	x e. lmems 0 -> x e. A
5	4	ax_gen	A. x (x e. lmems 0 -> x e. A)
6	5	conv all, subset	all A 0

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)