Theorem b0leb1 ≪ | index | src | ≫

theorem b0leb1 (a b: nat): $ b0 a <= b1 b <-> a <= b $;

Step	Hyp	Ref	Expression
1		bitr4	(b0 a <= b1 b <-> ~b1 b < b0 a) -> (a <= b <-> ~b1 b < b0 a) -> (b0 a <= b1 b <-> a <= b)
2		lenlt	b0 a <= b1 b <-> ~b1 b < b0 a
3	1, 2	ax_mp	(a <= b <-> ~b1 b < b0 a) -> (b0 a <= b1 b <-> a <= b)
4		bitr4	(a <= b <-> ~b < a) -> (~b1 b < b0 a <-> ~b < a) -> (a <= b <-> ~b1 b < b0 a)
5		lenlt	a <= b <-> ~b < a
6	4, 5	ax_mp	(~b1 b < b0 a <-> ~b < a) -> (a <= b <-> ~b1 b < b0 a)
7		noteq	(b1 b < b0 a <-> b < a) -> (~b1 b < b0 a <-> ~b < a)
8		b1ltb0	b1 b < b0 a <-> b < a
9	7, 8	ax_mp	~b1 b < b0 a <-> ~b < a
10	6, 9	ax_mp	a <= b <-> ~b1 b < b0 a
11	3, 10	ax_mp	b0 a <= b1 b <-> a <= b

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_peano (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)