Theorem b1leb0 ≪ | index | src | ≫

theorem b1leb0 (a b: nat): $ b1 a <= b0 b <-> a < b $;

Step	Hyp	Ref	Expression
1		bitr4	(b1 a <= b0 b <-> ~b0 b < b1 a) -> (a < b <-> ~b0 b < b1 a) -> (b1 a <= b0 b <-> a < b)
2		lenlt	b1 a <= b0 b <-> ~b0 b < b1 a
3	1, 2	ax_mp	(a < b <-> ~b0 b < b1 a) -> (b1 a <= b0 b <-> a < b)
4		bitr4	(a < b <-> ~b <= a) -> (~b0 b < b1 a <-> ~b <= a) -> (a < b <-> ~b0 b < b1 a)
5		ltnle	a < b <-> ~b <= a
6	4, 5	ax_mp	(~b0 b < b1 a <-> ~b <= a) -> (a < b <-> ~b0 b < b1 a)
7		noteq	(b0 b < b1 a <-> b <= a) -> (~b0 b < b1 a <-> ~b <= a)
8		b0ltb1	b0 b < b1 a <-> b <= a
9	7, 8	ax_mp	~b0 b < b1 a <-> ~b <= a
10	6, 9	ax_mp	a < b <-> ~b0 b < b1 a
11	3, 10	ax_mp	b1 a <= b0 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)