Theorem ltcons1 ≪ | index | src | ≫

theorem ltcons1 (a b c: nat): $ a < b <-> a : c < b : c $;

Step	Hyp	Ref	Expression
1		bitr	(a < b <-> a, c < b, c) -> (a, c < b, c <-> a : c < b : c) -> (a < b <-> a : c < b : c)
2		ltpr1	a < b <-> a, c < b, c
3	1, 2	ax_mp	(a, c < b, c <-> a : c < b : c) -> (a < b <-> a : c < b : c)
4		ltsuc	a, c < b, c <-> suc (a, c) < suc (b, c)
5	4	conv cons	a, c < b, c <-> a : c < b : c
6	3, 5	ax_mp	a < b <-> 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, muleq, add0, addS, mul0, mulS)