Theorem xp01 ≪ | index | src | ≫

theorem xp01 (A: set): $ Xp 0 A == 0 $;

Step	Hyp	Ref	Expression
1		eq0al	Xp 0 A == 0 <-> A. x ~x e. Xp 0 A
2		xpfst	x e. Xp 0 A -> fst x e. 0
3		el02	~fst x e. 0
4	2, 3	mt	~x e. Xp 0 A
5	4	ax_gen	A. x ~x e. Xp 0 A
6	1, 5	mpbir	Xp 0 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)