Theorem maxlt ≪ | index | src | ≫

theorem maxlt (a b c: nat): $ max a b < c <-> a < c /\ b < c $;

Step	Hyp	Ref	Expression
1		bitr	(max a b < c <-> max (suc a) (suc b) <= c) -> (max (suc a) (suc b) <= c <-> a < c /\ b < c) -> (max a b < c <-> a < c /\ b < c)
2		leeq1	suc (max a b) = max (suc a) (suc b) -> (suc (max a b) <= c <-> max (suc a) (suc b) <= c)
3	2	conv lt	suc (max a b) = max (suc a) (suc b) -> (max a b < c <-> max (suc a) (suc b) <= c)
4		maxS	suc (max a b) = max (suc a) (suc b)
5	3, 4	ax_mp	max a b < c <-> max (suc a) (suc b) <= c
6	1, 5	ax_mp	(max (suc a) (suc b) <= c <-> a < c /\ b < c) -> (max a b < c <-> a < c /\ b < c)
7		maxle	max (suc a) (suc b) <= c <-> suc a <= c /\ suc b <= c
8	7	conv lt	max (suc a) (suc b) <= c <-> a < c /\ b < c
9	6, 8	ax_mp	max a b < c <-> a < c /\ b < c

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)