Theorem sub01 ≪ | index | src | ≫

theorem sub01 (a: nat): $ 0 - a = 0 $;

Step	Hyp	Ref	Expression
1		le02	a <= 0 <-> a = 0
2		sub02	0 - 0 = 0
3		subeq2	a = 0 -> 0 - a = 0 - 0
4	2, 3	syl6eq	a = 0 -> 0 - a = 0
5	1, 4	sylbi	a <= 0 -> 0 - a = 0
6		nlesubeq0	~a <= 0 -> 0 - a = 0
7	5, 6	cases	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, add0, addS)