Theorem nlesubeq0 ≪ | index | src | ≫

theorem nlesubeq0 (a b: nat): $ ~b <= a -> a - b = 0 $;

Step	Hyp	Ref	Expression
1		leaddid1	b <= b + x
2		leeq2	b + x = a -> (b <= b + x <-> b <= a)
3	1, 2	mpbii	b + x = a -> b <= a
4		absurd	~b <= a -> b <= a -> x = 0
5	3, 4	syl5	~b <= a -> b + x = a -> x = 0
6	5	eqthe0abd	~b <= a -> the {x \| b + x = a} = 0
7	6	conv sub	~b <= a -> a - b = 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 (addeq)